The Teessiders will hold talks with the England B international in the next few days and Blades manager Bryan Robson is already resigned to the departure of the club's prize asset.
Everton, Wigan Athletic and Birmingham City are also interested and Jagielka has told the Blades that he does not want to play in the Championship next season.
Middlesbrough manager Gareth Southgate is desperate for a new right-back to replace Abel Xavier, who has joined LA Galaxy, and Andrew Davies who has been playing out of position.
Southgate was linked with Borussia Dortmund's Phillipp Degen but the capture of Jagielka a player on the brink of the full England squad would help pacify those supporters unhappy over the departure of striker Mark Viduka to Newcastle United.
At 24, Jagielka is keen to get rid of the 'utility player' label and hold down a regular position in a Premiership side.
A future as an England right-back is on the horizon after an impressive performance in the role against Albania's B side two weeks ago.
Last night, Robson admitted that uncertainty over the Blades' league status was not helping his attempt to find a replacement for Jagielka.
With only 10 days to go before the Premier League's handling of the Carlos Tevez affair is examined by independent arbitrators, it would be far from advisable to rush into the transfer market at this stage.
"We cannot go in making bids when we don't know what division we will be playing in next season," said Robson. "We are having to hold back and hope that people we are interested in will be patient.
"It's a situation we have got to look at," he said. "We can't plan purely for the Championship because we might be playing in the Premiership. Equally, we can't plan for the Premiership because we might be in the Championship. The sooner the issue is resolved the better."
Jagielka's departure will leave a major hole in the Blades' armoury as the
player operates equally impressively in midfield or defence.
The Blades have released right-back Rob Kozluk and centre-back Chris Lucketti is also likely to depart in search of more regular first-team football.
Robson, however, is encouraged by the fact that with the notable expection of Jagielka none of the club's top players are seeking to jump ship.
Goalkeeper Paddy Kenny, midfielder Michael Tonge, and striker Rob Hulse have all confirmed a desire to stay and work under Robson and help the Blades bounce back from relegation.
"All the feelings I'm getting from all the lads so far is that they want to stay which is great news," Robson told the Yorkshire Post.
"I would have preferred to keep Phil Jagielka but I think his mind was already made up to move before I came here.
"Phil has made it known that he wants to stay in the Premiership and everyone knows that he has got a clause in his contract which allows him to speak to other clubs. There is a lot of interest and he is currently looking for the best option."
Hollywood actor and Blades director Sean Bean, meanwhile, is to lead a delegation of 100 supporters to the Houses of Parliament next week where the club's protest against the Premier League will gather momentum.
The Blades want to raise awareness of their fight ahead of the arbitration on June 18 and Bean says the key issue is one of 'fairness' and sending out the right message to youngsters.
"United has always been close to my heart. I'm a massive Blades fan and I'm proud to be associated with anything that might help us overturn this injustice," he said.
"Everyone knows the facts, but we have had to bear the brunt of the unfair decision to fine West Ham rather than dock the club points. They broke league
rules, and points, not a fine should have been the punishment. Instead, West Ham have ended up being rewarded, while Sheffield United, who played by the rules, face relegation.
"I hope that the arbitration panel will see the injustice and the message that 'rule-breaking pays' will send out to kids playing football. The panel now has the chance to right that wrong and I'll be doing everything I can to encourage the arbitrators to do just that."
Blades chairman Kevin McCabe said: "We're thrilled to have Sean on board. He's a great guy and a genuine voice of the many thousands of disappointed Blades fans.
"We've already had a lot of support from MPs, and 35 have signed an Early Day Motion in support of our position. We hope that with this campaign, everyone in football will realise that Sheffield United should be able to take its rightful place in the Premier League next season."