The Jambos only had 14 outfield players at training on Thursday and two of them are on development loans and cannot face the Buddies, while the likes of Craig Halkett were only taking part in light training.
Neilson said: "We have a number of players injured so we will give the majority of them until the last minute. Some of them did a wee bit of training (on Thursday), some of them will do some (on Friday), some will do a fitness test on Saturday morning."
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Hearts had five players go off injured during last Saturday's 2-2 draw with Dundee United – John Souttar, Nathaniel Atkinson, Alex Cochrane, Gary Mackay-Steven and Cammy Devlin.
Michael Smith definitely remains out with a back problem, while Josh Ginnelly and Peter Haring were among those who sat out training on Thursday.
Fears that Souttar might have suffered another Achilles injury have been allayed.
Neilson said: "It's an ankle problem but I'm not going to go through them all individually because I could be here for the next hour.
"At this moment in time we have 11 players carrying knocks. It won't be 11 that will be out on Saturday. I would expect to probably have six of them back.
"I know about 90 per cent of who will start the game, there are one or two who are a bit borderline.
"There will be ones that will be a day, there will be some that are two days, there will be some that are two weeks, there are some that will be three to four weeks, but we have a squad that can handle it."
The Hearts manager may have to utilise Under-18 players to fill the bench, but he added: "We will have a strong team. Whether we will have the depth of the squad we have had recently, as in nine on the bench, we will probably not get to that.
"We will probably have four or five on the bench, but the ones we will have will be of good quality.
"We have had a great record throughout the season of having very few injuries and all of a sudden we get a number within a few days. The majority of them are contact injuries so there is nothing you can do about it.
"We don't have a big squad but we have a high-quality squad, so we should be fine. Because we went with a small squad but a real quality squad, any one of the 18 first-team players can quite easily go into the starting XI and be a top-four team."
Hearts beat Celtic in a Saturday night kick-off at Tynecastle on the opening weekend of the league season and Neilson is relishing the occasion.
"We did a lot of Friday nights in the Championship and did one Saturday night," he said. "I like it, you get the atmosphere before the game, the floodlights on and the build-up.
"The more exposure we can get to the game by putting it on TV, the better.
"For me, under the floodlights at Tynecastle is one of the best places you can play. When they walk out on that turf the place will be rocking.
"A club of Hearts' stature, we want to be winning trophies. We want to be challenging at the top end of the league, we want to get into European football and we want to win things. The Scottish Cup is a massive opportunity."