The Bundesliga club are yet to officially respond to the bids, but fresh reports in Germany claim Sesko has reached verbal agreements with both clubs. If and when Leipzig agree on the deals, it will be down to Sesko to decide on whether he moves to St James' Park or Old Trafford.
It's been a dramatic 24 hours in the pursuit of the 22-year-old Slovenian.
Newcastle made a club record offer of €80m plus €10m on Monday after getting encouragement that the striker was interested in the move to St James' Park.
But Manchester United then made their move, submitting an offer of €75m plus €10m. It comes amid fresh claims from the Athletic that Manchester United believe the striker would favour a move to Old Trafford.
Sesko was left out of Leipzig’s squad for Saturday’s friendly against Atalanta and didn't train on Tuesday amid the developments and uncertainty.
Newcastle will be desperate to succeed in their pursuit of Sesko having missed out on several leading targets already this summer. How the Sesko situation now plays out will have also have a huge bearing on the future of striker Alexander Isak.
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Newcastle last week rejected a £110million bid from Liverpool for the Sweden international, but the Premier League champions are likely to return with an improved offer if and when the Magpies have a replacement in place.
Isak is again at Newcastle's training ground today having returned to the club's Benton base on Monday. The striker didn't travel with the squad for their pre-season trip to Asia and instead trained alone at his former club Real Sociedad.
Showdown talks were planned this week with head coach Eddie Howe, who fired a warning to the striker at the weekend when he said: "You have to earn the right to train with us.
"We are Newcastle United. The player has a responsibility here to be part of a team and part of a squad - you have to act in the right way. So that is also at play here."
Isak wants the Liverpool move this summer, but Newcastle won't entertain the prospect of selling their star striker until they have a replacement in place. They also won't budge on their valuation of the 25-year-old, who still has three years to run on his contract with the Magpies.
Meanwhile, Newcastle remain hopeful of striking a deal for Yoane Wissa, with Brentford boss Keith Andrews admitting uncertainty over the forward's future "is not ideal".
Wissa wants to move to Newcastle and returned home early from Brentford's pre-season camp in Portugal last month. He's played no part in any of the Bees' summer friendlies so far, but the Magpies haven't yet been able to come to an agreement with their Premier League rivals.
Andrews said: “The situation is he’s training at the training ground but I think it’s probably public knowledge at this stage that he left that training camp in Lisbon.
"It was probably the right decision for all parties in that we had to support him in what is a difficult time for him.
“We need to remember that it’s not just about the footballer, it’s about the human being and the person first. Again, public knowledge that there is some interest in Wissa so we have to respect that, respect him.
“I’ve got a really good relationship with Yoane from last season and that is still the case now but obviously, it’s not ideal. \[There is\] constant communication with Yoane around the situation.
“We always treat people with respect and again, when you look at the business we’ve done this season with the outgoings, it’s been done early.
"This is probably not wanted. It’s not an ideal scenario but what transfer window is.”