Southgate has been linked with Erik ten Hag's job at United.
"I am not a high-stakes man and the betting companies can set their odds, but there are no conversations that have been had with me," Southgate told ITV News.
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Southgate took over the national team in 2016, after coaching the U21 national team.
"I have one thing to focus on. It is to have as successful a tournament with England as possible. Everything else outside is completely irrelevant to me."
Southgate previously managed Middlesbrough, but he has no plans to return to club football.
"If I start talking about all that, I'm already distracted from what I'm doing. I think the country expects me to have my focus on England, and I really am."