The Chelsea job could be up for grabs amid growing reports Sarri is set to sign a three-year deal to replace Massimiliano Allegri at Juve.
Southgate, who is contracted until 2022 after England's run to the 2018 FIFA World Cup semi-finals, has emerged as a possible replacement in the scenario that former Napoli boss Sarri leaves the Premier League side.
The 48-year-old, however, is not interested in returning to club football, having last managed Middlesbrough in 2009, as England prepares for Monday's UEFA Nations League finals third-place play-off against Switzerland.
"No," Southgate replied when asked if he would be interested in the Chelsea job. "I'm committed to England, simple as that.
"I don't need to hedge my bets in keeping doors open or keeping things alive. There's no way that I would leave England at this moment in time, to take any job.
"The only thing on my mind is that we've got a European Championship next summer and we've got to keep improving this team to be the best possible team we can be for the country.
"We brought 20,000 fans to Portugal which is remarkable, a sign of their engagement with the team and belief in what we're trying to do. So why would I rush to leave that opportunity and chance to affect not just the football but the other parts that the role brings? My sole focus is producing the best team possible for England."