The winger was a doubt following an injury sustained in the 1-0 win over Arbroath on Friday night.
Ginnelly was replaced in the first 10 minutes with a thigh complaint and left the stadium on crutches.
Neilson, whose side played a bounce game with Airdrieonians on Tuesday, is hoping he could be available, while there is also good news around Jordan Roberts who missed the league game at Gayfield after picking up a knock in training.
“The injury situation is alright,” he said. “We’re going to try Gino on Thursday to see if he’s ready. He left the stadium on crutches as a precaution on Friday but it settled down over the weekend. He’s desperate to be with us.
“Jordan Roberts should be fine, Elliot Frear will be fine and we had the majority of the squad playing in a bounce game on Tuesday afternoon to get them all ready for Saturday.”
Souttar 'miles ahead’
There was more positive news around John Souttar’s recovery from his Achilles injury with the player “miles ahead” of where the club expected him to be.
The centre-back suffered a relapse of the injury suffered against Rangers in February during lockdown. He was expected to be out until next season but Neilson isn’t ruling out a return later this campaign.
"John’s actually just out of his protective boot and is miles ahead of where we thought he’d be,” Neilson said.
"The plan is still next season for him, but you never know, he might be back earlier.
“We weren’t expecting the area to be in such good condition when the boot came off. He went back down to see the surgeon and he was really pleased with it.
"Now he’s stepped things up. He’s wearing trainers, has started on the bike.
"We didn’t want to put a time frame on things, which is why we said it would be next season before he’s back – but the more he goes and more he progresses, we might see him back a little earlier.”