The Argentine tactician has seen his profile rise considerably during a four-year stint in north London, with Spurs transformed into top-four regulars and Premier League title contenders.
Those efforts saw Pochettino touted as a possible successor to Zinedine Zidane at Real Madrid this summer, but he had already committed to a new deal in England.
Jenas believes the Blancos would appeal to the South American if they came calling again, while it would be difficult to knock back an approach from Old Trafford if Jose Mourinho was to move on.
The former Spurs midfielder told the Daily Star: “I think they’ve been after him for a long time to be honest with you. It’d be a huge test. I think deep down that’s the one Pochettino wants.
“There’s only United and Real Madrid that I think really take his fancy.
“There’s just two clubs really. I know you’ve got Bayern, PSG and so on but I think he’d prefer to stay in the Premier League or go back to La Liga.
“If that did happen [Mourinho leaves Man Utd], it would not surprise me if United go really hard for Pochettino. And, given what’s happened at Spurs, I’d be surprised if he stayed.”
While tipping Pochettino to leave Spurs at some stage in the near future, Jenas admits walking away with no silverware would be difficult for a man who has worked so hard to transform the fortunes of a success-starved club.
He added: “The only sticking point for United is he’s invested so many years at Tottenham.
“I always say to people the way he’s loved by players, handles the football club and handles himself - you don’t see him moan too much - I reckon all the biggest clubs in the world want Pochettino to manage them.
“It’s on his terms when he wants to push that button.
“It’s whether he feels there’s a rush and a need in his career to go down that road.
“I’m almost certain if he left Tottenham without winning anything, it would leave a horrible taste in his mouth.
“He’d be thinking: ‘Yes, I had a good time, but what did I actually do there?’
“The answer would be he didn’t win a thing.
“It would surprise me to see him leave without winning anything because Manchester United and Real are always going to be there whenever he wants the job.”