The Irons manager also believes that his team still have more to give following their surprise top-four challenge in a topsy-turvy Premier League term
David Moyes has admitted that he would be open to signing Jesse Lingard on a full-time basis from Manchester United if he can maintain the trajectory of his impressive rebirth at West Ham.
The former England winger scored twice on a glittering debut performance for the Hammers in their midweek 3-1 win over Aston Villa following his loan arrival from Old Trafford.
The 28-year-old reunited with his former Red Devils boss at London Stadium, starting his first Premier League game for over a year on Wednesday night.
While current United manager Ole Gunnar Solskjaer has claimed that the playmaker still has a future with the club, Lingard has cut an increasingly isolate figure at the Theatre of Dreams under the Norwegian.
But Moyes belives that the ex-Three Lions man could instead find greater purchase in the capital, where he could even push for a recall to Gareth Southgate's plans ahead of Euro 2020.
"He is a more mature boy now and is someone who is getting to the prime of his career," the Irons boss stated ahead of his side's clash with Fulham this weekend.
"He's at the stage where he wants to play well and he wants to be on the pitch. We're not getting carried away but we are obviously really pleased with Jesse.
"He was always a fit young player so getting some match practice for him is important. We don't want to put too much pressure on him but he has set himself a great standard by his performance.
"We've had one game. The situation here is Jesse's going to get a good chance to look at West Ham, see what he thinks of us and whether he likes it, and we'll get a good look at Jesse and decide what we think in five or six months' time."
In a particularly topsy-turvy Premier League season that looks to be finally coalescing into a familiar title race between Manchester City, Liverpool and United, West Ham have exceeded expectations to sit on the edge of a top-four challenge.
But Moyes, who guided Everton to a career-best fourth-place finish in 2005, feels that his side still have more to give, adding: "The position we are in is excellent.
"I still think we can do more. I don't think the players have quite reached where they have got to get to.
"We have shown good form in midweek after a couple of games where we had not been quite so good.
"So I think there is room to improve, and if we can improve in the second half of the season then hopefully that will keep us near the top end of the league."