Factbox - UK PM May's Brexit 'plan B': What happens next in parliament?

By Kylie MacLellan and William James

LONDON (Reuters) - British Prime Minister Theresa May is seeking to break the parliamentary deadlock over Brexit by proposing to seek further concessions from the European Union on a plan to prevent customs checks on the Irish border.

On Jan. 29, parliament will debate May's proposed next steps as well as alternative plans put forward by lawmakers, including some that seek to delay Britain's March 29 exit by requesting an extension to the two-year Article 50 negotiation period.

Some also seek to shift control of the process away from government and give parliament the chance to define Brexit. If successful, this could have a profound effect, giving lawmakers who want to block, delay or renegotiate Brexit a possible legal route to do so.

Below is what is due to happen next:

JAN. 21-29: LAWMAKERS PROPOSE ALTERNATIVES

Lawmakers have begun proposing alternatives to May's next steps through a parliamentary device known as an amendment. Amendments will be selected on Jan. 29 by speaker John Bercow and can then be put to a vote.

Below are the amendments that have been put forward so far:

AMENDMENT A

Proposed by opposition Labour Party leader Jeremy Corbyn, it calls for parliament to consider alternative options to prevent Britain leaving without a deal, including seeking a permanent customs union with the EU and holding a second referendum.

This is unlikely to be approved as pro-EU Conservative lawmakers have indicated they will not support it.

The pro-EU Liberal Democrats have proposed a change to this amendment so that it would call only for parliament to vote on holding a second referendum and that remaining in the EU should be an option in that referendum.

AMENDMENT B

Put forward by a group of Labour lawmakers, this calls on the government to request an extension to the Article 50 deadline so that a 'Citizen's Assembly' of 250 people can be created to consider the way forward and make recommendations to parliament within 10 weeks of being set up.

AMENDMENT C

This has been put forward by Labour lawmaker Hilary Benn, who chairs parliament's Brexit select committee. It calls on the government to hold indicative votes on the following options:

1) Holding another vote in parliament on May's deal

2) Leaving with no deal on March 29

3) Calling on the government to renegotiate May's deal

4) Holding a second referendum

AMENDMENT D

Proposed by lawmakers from Labour, May's Conservative Party and the Liberal Democrats, this calls on the government to request an extension to the Article 50 deadline.

AMENDMENT E

Put forward by Labour lawmaker Yvette Cooper, it has a strong chance of succeeding as Labour's finance policy chief has said it is "highly likely" the party will back it and it is also supported by several of May's Conservative lawmakers.

It seeks to shift control of Brexit from May's government to parliament by demanding that on Feb. 5, the rule that government business takes precedence in parliament is overturned.

Providing it has the support of 10 lawmakers, from at least four political parties, it then makes time for a piece of legislation Cooper has proposed, which gives May until Feb. 26 to get a deal approved by parliament.

If the government fails to get a deal through by that date, parliament would be given a vote on asking the EU for a postponement of the Article 50 deadline to prevent Britain leaving without a deal on March 29. It proposes a nine-month extension, to Dec. 31.

AMENDMENT F

This has been proposed by Conservative lawmaker Dominic Grieve and has a chance of succeeding as it is supported by lawmakers from several parties.

It demands that, one day a week in February and March, the rule that government business takes precedence in parliament is overturned, giving lawmakers the opportunity to propose their own debates on Brexit.

AMENDMENT G

Proposed by an independent lawmaker and a member of May's Conservatives, this amendment calls for parliament to be given votes on a range of Brexit options including leaving without a deal, extending Article 50, leaving with a Canada-style or Norway-style relationship and holding another referendum.

AMENDMENT H

Put forward by Labour and Conservative lawmakers, it seeks to rule out a 'no deal' Brexit.

AMENDMENT I

Proposed by Conservative lawmaker Andrew Murrison and supported by a large number of Conservatives, it calls for Britain's exit deal with Brussels to be changed to add an expiry date to the Northern Irish backstop of Dec. 31, 2021.

AMENDMENT J

Put forward by Conservative lawmaker John Baron, this amendment calls on parliament to reject any Brexit deal which includes a Northern Ireland backstop.

AMENDMENT K

Proposed by Liberal Democrat leader Vince Cable and his lawmakers, it calls on the government to rule out a 'no deal' exit and prepare for a second referendum in which the option to remain in the EU would be on the ballot paper.

AMENDMENT L

Proposed by Liberal Democrat lawmaker Tom Brake demands that a committee of no more than 17 lawmakers from across political parties be created and given control of the parliamentary Brexit process.

AMENDMENT M

Proposed by Conservative John Baron, it states that parliament will not approve a Brexit deal which includes a Northern Ireland backstop lasting any longer than six months.

AMENDMENT N

Also put forward by Baron, it calls for parliament to reject any Brexit deal that does not give Britain a unilateral right to terminate the Northern Ireland backstop.

JAN. 29: PARLIAMENT DEBATES AND VOTES ON NEXT STEPS

Parliament will hold a day of debate on May's proposed next steps and lawmakers' amendments. They will not be asked to vote to approve a revised Brexit deal at this stage.

A vote in favour of changing the parliamentary rules would change the long-held principle of the British parliament that the government has control of what has the chance to become law.

Votes on alternative types of deal proposed by lawmakers should give an indication of whether there is any way forward supported by a majority in parliament.

If an option were approved by a majority of lawmakers, May could go back to the EU and seek changes to her Brexit deal. Parliament would ultimately need to vote on any revised deal.

(Reporting by Kylie MacLellan, William James and Elizabeth Piper; Editing by Janet Lawrence)