Mawson has been one of the few Swans players to have emerged from the last couple of years with any credit amid the doom and gloom that has existed at the Liberty Stadium in recent times, and unsurprisingly, he is attracting plenty of attention from one of their Premier League rivals.
The Sun reports that Moyes wanted to bring the 23-year-old to the Stadium of Light when he was Sunderland boss, and he has now reignited that interest at the London Stadium - but the move now appears to be in some doubt.
According to The Sun, West Ham have been told that they will need to pay £40 million for the Englishman, and some sort of swap deal involving Andre Ayew may be required to help a move take place - though it seems unlikely at this stage.
The Sun adds that West Ham were previously in discussions with Swansea about selling Ayew back to them, but their desire to recoup the full £20.5 million that they paid for him, and Swansea's reluctance to pay that amount, has meant that there is still a long way to go before a deal may start to look possible.
Even in today's game, paying £40 million for a 23-year-old who only started playing in the Premier League in 2016 is surely a massive gamble, and it seems unlikely that West Ham will be willing to take such a risk next month when they are in such a precarious position in the top flight.
But if they were able to lower the price significantly by engineering some sort of swap deal involving Ayew, it would not be a surprise if a number of Irons fans were keen to see the club push for a deal.