Arsenal are currently on the hunt for a new head coach after the Gunners parted company with Unai Emery last week following a miserable run of form in all competitions.
The Gunners sacked Emery on Friday morning and are now looking for a new manager, with Freddie Ljungberg taking temporary charge at The Emirates.
Ole Gunnar Solskjaer has also found himself under pressure at Old Trafford this season, with the Red Devils having won just four of their opening 13 games in the Premier League this season.
Rodgers has been earning lots of praise for the impact he has had at Leicester City this season.
And former Arsenal star Nicholas feels that the Gunners should be considering him as a candidate for their next boss – as well as Manchester United.
Writing in his column for Sky Sports, Nicholas said: “If Man United do not see off Villa, and with games against Tottenham and Manchester City coming up, Ole Gunnar Solskjaer could be in trouble and I think they should go for Brendan Rodgers too, along with Arsenal.
“Mauricio Pochettino is floating everyone’s boat, but if you look at the real situation of how it has developed, then Rodgers is a better fit for both of those teams than Pochettino is, because he knows how to win trophies.
“He won nine domestic titles at Celtic and it might be a lot easier there but he knows how to do it.
“It is not a case of being insecure, Rodgers has his beliefs and strengths and goes and does it. I would be amazed if Manchester United were not interested in him.”
Arsenal will travel to Norwich City in the Premier League on Sunday afternoon, while Manchester United will take on Aston Villa at Old Trafford.
Both clubs have struggled to find consistent form in the top flight this season and both are considerably off the pace in the race for Champions League qualification.
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