The clubs will continue talks this weekend with the aim of reaching a compromise deal.
This comes after an opening bid of £23m was rejected, with the Italian club holding out for a transfer fee of around £35m.
Dorgu joined Leece in 2022 from Danish FC Nordsjælland, to the tune of just £168,000 [€200,000].
Initially joining the Serie A side's Under-19s squad, he spent just one season with the youth team before being promoted to the senior squad and made his first-team Lecce debut in August 2023.
In the one-and-a-half years since his senior debut, the 20-year-old has made 56 appearances, scoring five goals and providing two assists.
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His performances have also seen him rewarded with four caps for the senior Denmark national team.
And Lecce are now looking to make huge profit off the starlet, as they play hardball over a £35m valuation of a player they bought for less than £200,000 three years ago.
United head coach Ruben Amorim is aiming to land his first transfer since his appointment in November 2024.
As Man United sit 13th in the Premier League, it is clear the squad needs new additions to revive their season.
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Following an expensive summer window bringing in Leny Yoro [£58.9m] and Matthijs de Ligt [£42.9m], owners Ineos are much more reserved this time around and are looking to offload the likes of Antony and Marcus Rashford, who are on big wages, before splashing the cash.
But to secure Dorgu's signature, Man United must do so this window otherwise other interested parties may return.
Chelsea have registered their interest previously and the Dane is a self proclaimed fan of the Blues who would consider it 'a dream' to play at Stamford Bridge.
At Sporting Lisbon Amorim often played a 3-4-3 or 3-4-2-1 where the wing-backs are arguably the most crucial players.
Tasked with stretching the pitch by being high and wide, it allows for space to be created inside.
Not only that, they need to function as an attacking weapon able to deliver crosses and directly contribute to goals.
Dorgu as someone who has played multiple games as a left-sided wing-back and in theory fills Amorim's desire.
Diogo Dalot has often played in that position despite not being either a natural attacker or left footed player.
Tyrell Malacia since his return from injury has featured there occasionally while Luke Shaw continues to be sidelined with a muscular issue.
Fans will grow impatient if results do not turn around but Dorgu could be the start of a new chapter of Amorim ball.
It was a system that took Manchester City by storm when they lost 4-1 to Sporting Lisbon in the Champions League earlier this season with Amorim at the helm.