The 19-year-old is understood to be on United's list of left-back options this summer as Michael Carrick's side look to strengthen the position, with reports suggesting he is being considered as competition for Luke Shaw.
Lewis-Skelly struggled for regular game time in the first half of last season after emerging as one of Arsenal's breakout stars, with Riccardo Calafiori often preferred in his favoured left-back role, but has been urged by Wilshere, formerly Arsenal's Under-18s coach, to remain at the Emirates Stadium and fight for his spot.
Speaking to Sky Sports, Luton Town manager Wilshere said: "Last year was interesting to watch for Myles because I think he probably never experienced that. All the way through the academy, him and Ethan Nwaneri were probably always the best players in their age group. The academy have done a really good job of giving him the right challenge at the right time, but then all of a sudden he faced a bit of adversity. I was really happy for Myles with the way he managed it.
"Forget the ability and all of that, the one thing that I love about Myles is his character and personality and he's never let that leave him. When I'd watch on the TV, it would sometimes go to Myles on the bench and he would still cheer every moment. He's a character who wants to feel love and he gives a lot of love as well.
"He obviously found his way at the end of the season playing in the Champions League final, in my opinion, in his best position. I'd like to see him stay and fight for it because I believe he's good enough to find his way in that team. I believe he can do that."
Lewis-Skelly started in Arsenal's Champions League final last season and has previously featured for England at senior level, having come through the academy alongside Nwaneri as one of the standout talents of his generation.
He broke into Mikel Arteta's first team playing at left-back, despite predominantly playing midfield throughout his academy career, and quickly became a fans' favourite for his combative, front-foot style. His breakthrough form earned him a senior England call-up from Thomas Tuchel, and heavyweight interest from elsewhere, with Real Madrid previously credited with tracking his progress before Arsenal moved to tie him down to a long-term contract.
Arsenal's pre-season continues with friendlies against Real Betis, Borussia Dortmund and Como, before a Community Shield meeting with Manchester City on August 16 and the start of their Premier League title defence against newly promoted Coventry City five days later, with Lewis-Skelly's future expected to be resolved before then.
Should Manchester United press ahead with concrete interest, it would represent a significant statement of intent from Old Trafford, who are understood to view a long-term successor to Luke Shaw as a priority before the window closes.
For now, though, Wilshere's message is clear: Lewis-Skelly's best route back into regular football lies in staying and proving his case at the club that developed him.