The Scotsman was brought back to east London for a second spell in December and has plans to evolve the squad should he be given the chance in the long term, having only lasted six months despite keeping the club up last time around.
Former Manchester United boss Moyes is keen to bring a new youthful core to the Hammers first team and move the club away from expensive, big-name signings such as Samir Nasri, Patrice Evra and Jack Wilshere.
Such deals have proved rather unsuccessful in recent years.
For his plans to be put in place, however, the 57-year-old must keep the Hammers in the Premier League this season. After consecutive defeats since the restart, West Ham remain above the bottom three on goal difference alone, however, with not one of the bottom five winning since the season resumed, their fate remains in their hands with Chelsea the visitors on Wednesday night.
The Hammers marked their 125th anniversary on Monday, and Moyes wants to add a new chapter to the history books.
"West Ham has got a fantastic history but what we want is new history," said Moyes. "And I want the chance to build new history, I want the chance to make new history, I want the chance to do something which gives us all a feeling of hope in the future.
"But the big thing to do at the moment is to win the next four or five games to make sure that we are Premier League club. I think if we can do that, we can start building new history.
"And I want to be the guy to do it and I want to be the person here to do it. I've got a lot of thoughts and a lot of ideas where I want to take the club if I can do.
"But it would be completely wrong not to mention that there have been some unbelievable managers at this club, some incredible players who would be recognised worldwide. So somewhere in the future we have got to believe we can do that again."