The Red Devils have officially parted ways with head coach Erik ten Hag and are now in search of a new permanent head coach.
A number of different names have been linked to the role, including Howe, who is currently a 25/1 shot according to bookmakers Sky Bet.
Barnes has told Howe to think twice about swapping Newcastle for Manchester, though, after being told by Lucky Block that United were ‘keeping tabs’ on the Magpies boss.
‘Manchester United are a bigger club than Newcastle, but if he went there, would things work out better than they have for him at Newcastle?’ Barnes said.
‘At least the Newcastle players will play for him, and I’m not sure you could say with any confidence that that would be the case at Old Trafford.
‘Will he turn down Man United if they come calling? I think that Eddie would be jumping straight out of the frying pan and into the fire with that if he were to join them.
‘That’s not to say he won’t do it, because the Manchester United job is something that should appeal to all ambitious managers, but there are obviously at lot of issues at the club at this moment in time.’
Howe, regarded as one of the best active English managers in football, has been in charge of Newcastle for three years and has shined in his role at St James’ Park.
The 46-year-old took the Magpies job in November 2021 – with the team sat inside the Premier League relegation zone – and helped them finish up in 11th.
In the following 2022-23 season, his first full campaign, Howe impressively led Saudi-backed Newcastle to a fourth-placed finish to qualify for the Champions League and they were also runners-up in the Carabao Cup to Erik ten Hag’s side.
Although they had somewhat of a dip last term, finishing seventh, Howe’s side were rocked by a number of key injuries. They look set to challenge for Champions League qualification again this season, although they are currently six points off the top four following a slow start.
Howe holds legendary status at his former club AFC Bournemouth, having guided them from the bottom of League Two – on minus 17 points – all the way up to the Premier League for the first time in their entire history over two managerial spells.