Southgate's current England contract expires in December and he is yet to agree a new deal ahead of his fourth major tournament with the national team.
The 53-year-old is reportedly United's top target to replace the under-fire Ten Hag if Old Trafford co-owner Jim Ratcliffe decided to sack the Dutchman.
Southgate on Friday said contract talks with the Football Association are on hold, with his full focus on leading England to their first men's international trophy since the 1966 World Cup.
"I think there are two things from my point of view. One is that I'm the England manager. I've got one job, basically, to try and deliver a European Championship," Southgate said ahead of Saturday's friendly against Brazil at Wembley.
"And the second thing is Manchester United have a manager and I think it's always completely disrespectful when there's any speculation about a manager that's in place.
"I'm president of the LMA (League Managers' Association) so I don't have any time for that sort of thing, really."
Southgate is said to have a good relationship with Dave Brailsford, who is part of Ratcliffe's football operations team at United.
He is also on good terms with Newcastle sporting director Dan Ashworth -- another potential candidate to move to Old Trafford according to reports.
Southgate's stock is high after leading England to the 2018 World Cup semi-finals, the European Championship final in 2021 and the World Cup quarter-finals in Qatar a year later.
Irrespective of whether the chance to join United eventually comes up, Southgate said he will not speak to anyone until after the Euros.
"I think I've answered this every time I've sat with you," Southgate said. "Yeah, my focus is a European Championship.
"If we did something, a contract, here before everybody would be saying, 'why are you signing a contract before a Euros where you've got to prove yourself?'.
"I'm certainly not going to speak to anybody else ahead of that. I never have.
"I've been eight years in the job. I wouldn't entertain speaking to anybody else when I'm in the job."
