The report states that Palace boss Alan Pardew faces a tough battle in keeping hold of the goalkeeper this summer as Everton could offer him the chance to follow in the footsteps of boyhood idol Neville Southall.
The 29-year-old has been a key figure for the Eagles this season, helping the club turn the corner after a poor league run while also playing a pivotal role in their impressive run to the FA Cup final.
With only one more year left on his contract, the Welsh stopper could be available for a bargain price this summer and there are a number of clubs looking to strengthen in that area.
Everton, set to lose Tim Howard in the summer, are certainly among those in need of a reliable, top class custodian and Hennessey may be one of the best out there available at a decent price.
The Toffees have struggled in defence all season, with an inconsistent back five largely to blame.
Bournemouth are also another one who will look for an upgrade on the ageing Artur Boruc while Boro, should they earn promotion, may look to bring in a number one to replace 37-year-old Dimi Konstantopoulos.
The report names Everton as the club leading the chase for Hennessey, although between now and the end of the season there could be a number of other sides monitoring his situation.