Barcelona signed De Jong from Ajax in 2019 for £65m plus add-ons and extended his contract the following year, tying him to the club until 2026
However, club sources say the deal is not done and there are still significant issues to work through.
But the development is seen as positive in terms of the transfer eventually reaching a conclusion.
The 25-year-old De Jong is new United manager Erik ten Hag's number one summer transfer target.
United are also battling with Lyon to sign Feyenoord's Netherlands defender Tyrell Malacia, who is valued at 15m euros (£12.9m).
The 22-year-old left-back, who has five Dutch caps, played the full 90 minutes in both recent Nations League victories over Wales and was part of the Feyenoord team beaten by Roma in last month's Europa Conference League final.
Lyon thought they were well placed to complete the transfer, but now United have stepped in and made their interest known.
United boss Ten Hag had heavily scouted Malacia as Ajax manager
United already have Luke Shaw and Alex Telles battling for the left-back role but new manager Erik Ten Hag is an admirer of Malacia, who made his debut for Feyenoord when he was 18.
Ten Hag led training for the first time on Monday with his non-international players.
United are still to make any signings this summer as they await a response from Denmark midfielder Christian Eriksen, 30, who also has an offer on the table from Brentford.
Ten Hag and his players are due to leave for a pre-season tour of Thailand and Australia on 8 July.
Ideally, the De Jong deal would be completed by then, although there is no deadline being put on the transfer.
-- Courtesy of BBC Sport