Barcelona sporting director Deco has previously spoken of his admiration for Rashford – as well as Liverpool winger Luis Diaz.
“When we go to the market, there are names that we know and that could improve the team,” he told Catalan radio station RAC1 in May.
“We like Luis, we like Rashford and we like other players,” he said.
For his part, Rashford last month spoke of how he would like to play alongside teenage superstar Lamine Yamal at Barcelona.
“Everyone wants to play with the best. Hopefully… We’ll see,” he said in an interview with Spanish influencer Javi Ruiz.
Rashford’s Old Trafford contract runs until June 2028 but the England international has not been given a squad number for the new season and was left behind to train at Carrington while the first team travelled to Sweden, where they drew 0-0 with newly-promoted Leeds United in their first match of pre-season on Saturday.
Rashford came through the academy ranks at his hometown club, making his debut in February 2016 as an 18-year-old, under Louis van Gaal.
He has gone on to score 138 goals for the club in 426 appearances but has not pulled on a United shirt since being substituted during a 2-1 Europa League win over Czech side Viktoria Plzen on 12 December.
Amorim dropped Rashford for last season’s league derby against Manchester City at the Etihad later that month, going public with his concerns over the efforts he and team-mate Alejandro Garnacho had put into training.
If he were to complete a move to Barcelona, he would be joining a side that won a league and cup double in Hansi Flick’s first season as manager.