The 21-year-old is the Portuguese side's first-choice left-back and starred in a 4-0 win over minnows Feirense on Sunday night.
According to the Daily Mail, United are monitoring the ex-Barcelona youngster with the club currently plagued by troubles at the back.
Current left-back Luke Shaw was singled out for criticism by manager Jose Mourinho are struggling in United's 3-1 defeat to Watford two weeks ago.
Marcos Rojo toiled against Feyenoord in the Europa League last month while Daley Blind struggled as United drew 1-1 with Stoke this afternoon.
United have shipped in seven goals from their first seven league games - a far cry from last term where they conceded a joint-low 35 goals all season under old boss Louis van Gaal.
Manchester City are also reported to monitoring Grimaldo, with Pep Guardiola said to be unconvinced by Aleksander Kolarov and Gael Clichy.
The Benfica star has a buyout clause of £51million but they could be open to doing business for a reduced fee.
United have endured mixed fortunes over the years when signing players from the Portuguese top flight.
Cristiano Ronaldo and Nani were hits after signing from Sporting Lisbon, but Anderson proved to be a huge flop despite costing the club £30million to sign him from Porto in 2007.