Sunderland have issued a "hands-off" warning to Southampton as the Saints start the search for a new manager.
Mauricio Pochettino's departure from the St Mary's Stadium for Tottenham, which was confirmed on Tuesday evening, prompted a whirlwind of speculation over who is to succeed him, and Poyet's name was among those in the mix.
Murat Yakin and Ronald Koeman were being quoted as the bookmakers' favourites, but Poyet's odds were slashed overnight.
However, Press Association Sport understands the Wearside club will resist any attempt to lure the 46-year-old Uruguayan back to the South Coast, where he enjoyed a successful spell with Brighton before an acrimonious departure and his subsequent move to Sunderland, who he guided to Premier League safety.
Poyet's stock has risen markedly in recent months after he replaced Paolo Di Canio at the Stadium of Light with the club having collected just a single point from its first seven games prior to his arrival.
Remarkably he took them to the Capital One Cup final, where they were eventually beaten by Manchester City after taking a shock lead.
However, the Black Cats looked destined to return to the Football League when they found themselves seven points adrift at the foot of the table with just six games left to play, only to mount a remarkable fightback which saw them take seven points from trips to City, Chelsea and Manchester United on the way to a 14th-placed finish.
After the turmoil of the Di Canio era, Poyet's approach has provided a breath of fresh air, although there remains an element of uncertainty over his future.
There was disquiet in the boardroom over his assertion that the perennial fight for survival might suggest that there was something wrong within the club beyond the powers of successive managers, and as he celebrated what he referred to as "the greatest escape", his future remained the main topic for debate.
Speaking after a final day defeat by Swansea, Poyet said: "Now I will have a meeting with the chairman because it's natural, natural because yes, I want to know where we are going, if it's possible not to make the same mistakes that we have made in the last couple of years because I don't wasn't to go through another year like this one.
"It's natural, it's natural. There are teams who start the season and the aim is to stay up, so 12th to 17th.
"Are we that team? I don't know. Or are we gong to go for the top 10 in a better way?"
At the same time, Poyet, who expressed his admiration for the Saints last season, insisted he expected still to be in charge at the start of the new campaign, and he and director of football Lee Congerton are busily working on their transfer targets as they attempt to avoid the mistakes made 12 months ago.
It is understood there is little appetite for change either within the hierarchy and that owner Ellis Short is prepared to fight to hang on to a head coach who is understandably hugely popular with the fans.
Despite the speculation, it was business as usual for Poyet and Congerton on Wednesday as they completed their first summer signing by landing out-of-contract West Brom defender Billy Jones on a four-year-deal.
Significantly, the 27-year-old, who will formally join the club on July 1, is British and has top-flight experience in this country, combined attributes none of last summer's 14 signings could boast.