The versatile midfielder’s contract at Manchester City expires this month and the club are believed to have offered him improved terms to stay.
But, at 29, he started just 18 of the 32 Premier League games he appeared in last term and he is understood to be interested in a move to a club that would guarantee him a more regular first-team place.
Sky sources understand Milner will not make a decision about his future until after the upcoming England games against Republic of Ireland, a friendly on June 7, and Slovenia, a crucial European Qualifier away from home on June 14.
Milner has been previously linked with a switch to Liverpool. It is understood a number of clubs from across Europe are interested in his signing and Arsenal are also one of them.
During limited appearances this term, Milner managed seven Premier League assists and scored five times.
He has played 198 times in all for City, scoring 18 goals, since joining from Aston Villa for £26million in 2010. He was a Leeds trainee before joining Newcastle for £3.6m in 2004, and has 53 caps for England.