United triggered a one-year option in Rashford's contract in December that will keep him at Old Trafford until 2024, but the club are keen to agree a long-term deal.
Sources close to Rashford insist that the striker is focused on having a successful end to the season but that representatives expect to make a step forward in negotiations before the end of the campaign.
United, according to sources, would like to reach an agreement by the end of the year and ideally by the start of next season.
French champions Paris Saint-Germain have shown interest in Rashford in the past, but the 25-year-old is expected to stay at United, where he is enjoying the best season of his career under Erik ten Hag.
United are looking for a striker ahead of the summer transfer window, but Rashford remains a key part of Ten Hag's plans.
Sources have told ESPN that the club are ready to listen to offers for a number of senior players in a bid to increase the summer budget and navigate financial fair play constraints.
Scott McTominay, who scored both goals in Scotland's 2-0 win over Spain on Tuesday, is wanted by Newcastle United, while there is also interest from the Premier League and Europe in Harry Maguire, Victor Lindelof and Dean Henderson.
Meanwhile, United are carrying out an internal investigation before making a decision on Mason Greenwood's future after criminal charges against the 21-year-old were dropped in February. Greenwood is under contract until 2025 but hasn't played or trained since January 2022.