Manchester United have increased their offer for Bruno Fernandes twice in the past 24 hours, Sky Sports reports.
It is believed the second bid is worth 55m euros (£46.59m).
One of the proposed add-ons will see Sporting Lisbon receiving a further 10m euros, if Fernandes plays a certain number of games for United.
This comes amid reports that Barcelona are in talks with Sporting and have proposed signing him this summer until 2025.
However, the LaLiga champions want to loan him to Valencia until the end of the 2020-21 season.
Barca’s bid also appears to match Sporting’s valuation, so the decision on which club to join will rest with the player.