The young Belgian is set to move to Old Trafford for a fee of £16m plus bonuses, though late competition has arrived from Galatasaray and Inter Milan.
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Galatasaray made a bid of their own on Saturday night. However, the Mirror says Lammens' preferred choice is to join United this week.
Any deal for Lammens will cast doubt over the future of United's senior keepers Andre Onana and Altay Bayindir.
It's been suggested manager Ruben Amorim has lost confidence in Onana and is prepared to see him moved on before next week's transfer deadline.