In the final year of his contract, Walker’s second spell at Tynecastle looks almost certain to be coming to an end with the attacking midfielder having not featured since a substitute appearance against Ross County on September 18.
St Johnstone are one club looking into the possibility of signing the 28-year-old. Livingston have also been interested in taking him.
Walker will be free to discuss pre-contract terms with other team, though Hearts would have to sanction any deal which will take him away from Edinburgh prior to the conclusion of the season.
Still a big fan of Walker’s talents, Neilson insists any deal agreed in January which allows Walker to leave will have to suit Hearts as well as everyone else involved.
"He’s an experienced player and he’s out of contract in the summer. His agent is speaking to a few clubs,” said the head coach.
“I really like him, I think he’s a top player. But I understand he’ll want to play as he’s not had the game time he would have liked this year. It’s one we’ll assess in January and see what’s best for the player and the club.
“He’s a player who can win football matches and there are not many of them about. So we’re reluctant to give him up unless it suits both parties.”
Though Neilson and sporting director Joe Savage have both spoken of their desire to add to the squad in the New Year, the manager doesn’t foresee a scenario similar to the 2015/16 campaign, where Don Cowie, John Souttar and Abiola Dauda were all brought in on transfer deadline day.
He does, however, hope to soon see a bit more activity with regards to new player contracts. While Hibs have been busy across town securing some of their biggest assets to long-term deals, Hearts have yet to tie down the likes of Stephen Kingsley, Craig Halkett, Craig Gordon, Peter Haring, and the aforementioned Souttar, all of whom will be out of contract in the summer.
He also had a word of warning for supporters who are desperate for the club to attract another “goalscorer” after seeing the team fail to find the back of the net against Rangers with leading marksman Liam Boyce having an off-day inside the penalty box.
“Right at the top of the January wish-list is another striker. Preferably one that scores goals – but they are the hardest to get in January,” he said. “We’ve got a couple of strikers here at the moment but we’re trying to get something that gives us a different option up front.
“We’re not going to make a lot of changes to the squad – it’ll be one or two. It’s about getting the deals done but I don’t think we’ll be running around at the last minute trying to bring three or four players in like previously here and at other clubs. I think it’ll be a calm one this year.
"Joe is grafting away as we speak [on new contracts] and hopefully he’s got a few of them locked in his office! Hopefully we’ll get some things done in the next week or two.”
For the meantime the focus is on rebounding from last weekend’s tough 2-0 defeat to Rangers in Gorgie. The home side had plenty of chances, both before an after the visitors netted a quick-fire double to ultimately win the game, but couldn’t find the killer touch. On paper, Saturday’s opponents Dundee appear to be an easier proposition with the Den’s Park side having lost three straight and sitting in tenth place, but in this season’s cinch Premiership there are few, if any, easy games.
“I would agree their performances have been better than results would indicate,” said Neilson. “I watched their game the other night against Hibs, and they had a couple of chances. Danny Mulllen had a good one and if he takes that it’s a different game and the own goal kills it.
“But when you look at their squad they’re pretty experienced in this league, they look a Premiership team. So from our perspective going up there is going to be tough.
"With the Rangers game we had a lot of possession in a lot of good areas, a lot of chances, we just didn’t take them on the day. That happens when you are playing against good teams, they had a couple of chances and took them. So we have to learn from that.
“It’s a different game up at Dens Park, that’s for sure. It’s a difficult place to go. The pitch, the environment, away from home, it will be a different way to try and win the game.”
Neilson is at least emboldened by the fact Hearts have bounced back with a win after each of the previous three defeats. Saturday’s match isn’t just about repeating the feat, it’s also about putting together a run of victories – they haven’t won consecutive games since late September/early October – to give themselves a tighter grasp on third and some momentum going into next month’s derby.
"I think this wee period is a difficult one, but it can be a fruitful one if we can put our foot to the metal and get going,” he added.
“We have four games in quick succession, we have an away game, two home games and then the derby. So we’re hoping to pick up as many points as we can and hopefully others will falter.
“We’ve got a good group here with a few leaders who are positive. It’s about making sure they are right and they can replicate that in the dressing room. It’s about bouncing back quickly."