United are hoping to sign Rabiot, 27, from Juventus in a £15million transfer.
Murtough even flew out to Turin to speak with the player's mother, who happens to be his agent.
Rabiot and mother and agent Veronique are holding out for £8.4m-per-year to leave Juve for United, says The Sun.
Should a move fail to materialise, the player could stay at Juve and see out the final year of his contract before commanding a high wage as a free agent next summer.
Murtough was pictured leaving Turin on Friday and the outcome of the talks over Rabiot are yet to be revealed.