The bookmakers' favourites to replace Jose Mourinho go head to head at Wembley on Sunday afternoon, when Red Devils caretaker boss Ole Gunnar Solskjaer (5/2) takes on front-runner Mauricio Pochettino's (11/8) Tottenham.
But Southgate dominated talk ahead of the Premier League encounter after the Sunday Telegraph reported that United are considering an approach for the Three Lions boss due to increasing concerns that Pochettino will stay put.
Press Association Sport understands no one has been ruled in or out at this stage as the club assesses their options ahead of naming Mourinho's permanent successor in the summer.
Pochettino is the favourite, and believed to be interested in the job, but for now United are continuing their extensive and thorough managerial search.
Southgate's place on the short list makes sense after his work at the England helm.
The 48-year-old led the country to the World Cup semi-finals for just the third time, before sealing the Three Lions' place at the inaugural Nations League finals.
Southgate has also impressed in his stint in charge of the England Under-21s, but his only previous job in club management was at Middlesbrough - a three-year spell that included relegation in 2009.
Asked last month by Sky Sports how hard it would be to turn down a big club job, the England manager said: "At the moment that's very easy because I don't have anything to consider.
"So, we've just had our best year in about 28 years, we're in another semi-final next summer and I signed a new contract three months ago.
"I'm incredibly proud to be the England manager and I have the opportunity to lead England into a European Championships, where so many of the matches are at Wembley. So, for me, it's a very simple situation at the moment."
Southgate is sure to be asked about the United job soon enough but may well remain silent, just as Tottenham boss Pochettino has until now.
Solskjaer, by contrast, has been explicit about his desire to stay on at the club he helped win the treble as a player in 1999.
The Norwegian has emulated Sir Matt Busby in winning his first five matches at the helm and is due to return to Molde in the summer.
Former Real Madrid boss Zinedine Zidane and Juventus manager Massimiliano Allegri are others to linked with the manager's job, while United are also working to bring in a head of football reporting to executive vice-chairman Ed Woodward.
