The 22-year-old was in impressive form for Porto during the 2021-22 campaign, scoring 21 goals and registering five assists in 46 appearances in all competitions.
Meanwhile, the Brazilian has found the back of the net on 25 occasions in 70 outings for his current side since making the move from Tombense in September 2020, in addition to providing five assists.
According to O Jogo, Man United recently made an offer of €60m (£51.6m) plus €5m (£4.3m) in add-ons for the forward, but the bid was quickly turned down by Porto, who have no intention to sell this summer.
However, the report claim that the 20-time English champions are now preparing a fresh offer, which is likely to be closer to €80m (£68.8m).
Evanilson scored 14 times and registered four assists in 30 league appearances during the 2021-22 campaign, but he failed to find the back of the net in eight outings in the Europa League and Champions League.
The attacker has made three appearances for Brazil Under-23s but is yet to feature for the senior side.
Brighton & Hove Albion and Wolverhampton Wanderers have also been credited with an interest, but the English duo cannot compete with Man United financially for the South American.
The Red Devils were strongly linked with Darwin Nunez before the Uruguay international moved to Liverpool, with the club short of options at centre-forward following Edinson Cavani's departure.
There have been suggestions that Cristiano Ronaldo could move on this summer, but the Portugal international expected to spend at least another season with the club.
Anthony Martial is also heading back to Old Trafford following a loan spell at Sevilla, but the 20-time English champions will allegedly attempt to sell the France international in the current market.
Marcus Rashford will reportedly be given the chance to reignite his Old Trafford career under Erik ten Hag, while Jadon Sancho will be looking to kick on after an underwhelming first campaign at the club.
Man United's summer budget is believed to be in the region of £130m, and the bulk of it could be spent on Barcelona midfielder Frenkie de Jong.
Christian Eriksen is also said to be a leading target for the Red Devils, with the Denmark international allegedly being offered a contract, but the Premier League side are set to miss out on Ajax defender Jurrien Timber.