The Hammers defender is seen as an indispensable part of the side by the club, but has been interesting teams including Liverpool.
West Ham United have enjoyed a fine season but are already looking ahead to greater things next term at the Olympic Stadium, and the Daily Mail reports that the club are keen to agree terms with Aaron Cresswell on a new contract sooner rather than later.
Cresswell has become arguably one of the club's most important players since his arrival from Ipswich Town in 2014, and was awarded a new and improved five-year deal last summer after a tremendous debut season.
In his first campaign at Upton Park he played in all 38 Premier League matches and each of the Hammers' four FA Cup ties, and was named both the supporters' and players' Player of the Year 12 months ago.
The 26-year-old has kicked on even further this season under Bilic, making 47 appearances in all competitions, and has now played the full 90 minutes in 75 of the 76 league matches the club have played since his arrival.
That superb form is thought to have attracted serious interest from clubs such as Manchester City and Liverpool, both of whom are in the market for a new left-back, but the Mail reports that co-owner David Sullivan is preparing a second new contract offer in the last year to ward off their circling rivals.
One major concern for the Hammers will be that Cresswell is a born and bred Merseysider and began his youth career at Anfield before leaving for Tranmere Rovers as a 14-year-old.
However, the club hope that the offer of another improvement on his terms, as well as their challenge for a top-six finish this season and the promise of playing in front of 60,000 fans at the Olympic Stadium next season, will convince their key man to remain in East London.