The 26-year-old is in the final year of the deal he signed when moving from Livingston in the summer of 2019 and will be able to discuss pre-contract terms with other clubs in the January window.
Neilson has already spoken of his desire to get John Souttar and Craig Gordon signed up beyond the summer, while it was recently revealed the Tynecastle club are looking to make Alex Cochrane’s stay from Brighton & Hove Albion permanent.
“We’re speaking to Craig Halkett as well,” Neilson said. “We’ll try and get something done. I’d love to keep him; he’s done very well this season for us. He’s improved as the team has improved.”
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“He played in centre half in a back two in Championship and did well, but playing in the centre of the three definitely suits him more.”
“We probably did miss him last week. I think the physicality, the way Motherwell play, being direct, strikers using their physicality backing in, getting into areas, because of that and gaining territory, so I think Craig would’ve helped us.”
Hearts face St Mirren in the cinch Premiership at Tynecastle tomorrow and Neilson revealed there are no fitness concerns ahead of the game.
He said: “We’re all good, everyone’s back, [Liam] Boyce is back in, Halkett is back in. They’re fully fit."