Despite having qualified for next season's Europa League, the Hammers are yet to make a first-team addition this summer, only confirming a permanent deal for loanee Craig Dawson.
However, Eurosport claims that Moyes is keen to hold discussions with Jones regarding a switch to the London Stadium.
With the 29-year-old not playing a competitive game since January 2020, largely due to injuries, United will give consideration to parting ways with the centre-back on a free transfer.
The report adds that West Ham would prefer to hand Jones an incentivised contract due to his prolonged spell on the sidelines.
While Jones could remain at Old Trafford, he would only be regarded as fifth choice at best with Raphael Varane due to arrive from Real Madrid.