The England international has entered the final 12 months of his current Arsenal deal to prompt speculation about his long-term future.
Oxlade-Chamberlain has been linked with a potential move to Chelsea and Liverpool in a £25m move but a transfer deal has failed to materlialise so far.
Arsenal manager Arsene Wenger is thought to be prepared to allow Oxlade-Chamberlain’s contract run down, throwing up the possibility of the midfielder leaving on a free transfer next summer.
Manchester United lost 2-1 to Champions League holders Real Madrid in the Uefa Super Cup on Tuesday night as Ander Herrera and Paul Pogba were swamped in midfield.
Former Republic of Ireland defender Cunningham was critical of the Spain international’s performance against Real Madrid in Macedonia.
And the ex-Wimbledon defender suggested that Manchester United should put together a lucrative offer to bring Oxlade-Chamberlain to Old Trafford from Arsenal to replace Herrera.
“Alex Oxlade-Chamberlain has a year left on his contract at Arsenal, they should be breaking the bank to bring that type of player to the club,” Cunningham told RTE Sport.
“That would improve them in that area of the pitch.”
Oxlade-Chamberlain scored two goals and made seven assists in 29 appearances in the Premier League last season.
Manchester United have signed Swedish defender Victor Lindelof, Everton striker Romelu Lukaku and Chelsea midfielder Nemanja Matic this summer.
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