The long-serving Arsenal star is embroiled in a contract stand-off with the club and has rejected a proposed wage cut.
Wilshere has threatened to quit if Arsenal, but the club has now told him they will not be making an improved offer.
The Mirror claim Liverpool and City are closely monitoring the situation with a view to a summer transfer.
Liverpool poached Alex Oxalde-Chameberlain from Arsenal last summer, while City have a long history of luring the club’s top stars to Manchester.
Wilshere is currently on international duty with England and wants his future resolved before the close of the season.
The Englishman has managed to fight his way back in Arsene Wenger’s side after spending last season on loan at Bournemouth.
Wenger told Wilshere he was free to leave the club last summer, but the 26-year-old decided to stay and battle for his right to earn a new contract.
Wilshere has already received offers from Serie A sides AC Milan and Juventus, as well Everton, but wants to remain at his boyhood club.
Arsenal 2017/18 fixtures
Stoke City (H) – Sun Apr 1
CSKA Moscow (Europa) (H) – Thu Apr 5
Southampton (H) – Sun Apr 8
CSKA Moscow (Europa) (A) – Thu Apr 12
Newcastle United (A) – Sun Apr 15
West Ham (H) – Mon Apr 23
Manchester United (A) – Sun Apr 29
‘We have known each other long enough where we can have that relationship where we are honest with each other,’ Wilshere said when asked about Wenger.
‘It was boiling up for a while because everybody knew I had a year left on my deal and I had been out on loan, got injured, and wasn’t really in his plans.
‘He said, ‘I am going to be honest with you and at the moment we are not going to be offering you a contract, so if you can get a contract somewhere else, you can go’.
‘Obviously I was not happy with that, but at the same time I was happy he was being honest. ‘He gave me the opportunity with three or four weeks left in the transfer window, but I did not find anything that I wanted and at the same time I was not really fit, so I decided that I wanted to stay and build up my fitness.
More: Arsenal FC