The 22-year-old Brazilian, who scored 21 goals for the Portuguese champions last season, is apparently on Erik ten Hag’s wishlist.
Signing a striker remains a priority for Ten Hag this summer and Manchester United have now shifted attention to Evanilson after previous target Darwin Nunez opted to join Liverpool.
Ten believes signing a new centre-forward would ease the pressure on 37-year-old Cristiano Ronaldo.
Ronaldo scored 24 times in the first season of his second spell at Old Trafford but departure of Edinson Cavani this summer means there is little back-up for the Portuguese star.
Manchester United intend to return with a new offer after Porto rejected the first offer of £55million.
The new, improved offer is expected to be close to £69million (€80m) and it is believed Porto would accept it.
Porto reportedly value Evanilson at £70million, a price that scared off fellow Premier League clubs Wolves and Brighton, who enquired after Evanilson.
Porto would like to keep hold of Evanilson, whose contract runs until 2025, but the report on Daily Mail claim an improved offer from Manchester United would test Porto’s resolve.
Evanilson has scored 25 goals and provided five assists in 70 appearances for Porto following his transfer from Brazilian club Fluminense in September 2020 for £8million.
The Fortaleza-born player emerged through the academy ranks at the Rio-based club and quickly attracted admirers in European football.
Meanwhile, Manchester United have suffered a huge blow in the pursuit of Ajax star Jurrien Timber.
Timber had reportedly been close to agreeing a transfer worth around £30million before he made a U-turn following a warning from Van Gaal.
Dutch journalist Marcel Van Der Kraan told Sky Sports: “Absolutely dead (the deal) in the water.
“The player was ready to come. The player’s agents were in London at Manchester United’s offices two weeks ago.
“The talks were very good, money was never a problem but then Louis van Gaal turned up and told the player in the Dutch camp ‘if you go there I think your chances could be limited for playing for Holland in the World Cup’.”