The stand-in skipper has been training away from the squad this week after feeling his hamstring tighten in the defeat to Villa at Turf Moor.
Tarkowski, 29, is a major doubt for the penultimate away game of the season at the Tottenham Hotspur Stadium, but April's Premier League Manager of the Month Jackson is hopeful he'll return before the end of the season.
"We're still waiting on Tarky, I wouldn't push or risk anybody if they were injured, if the players are right to play and fit enough then they will be up for selection.
"They're just managing it, they've not really looked at it properly in terms of that [a scan]. We'll have to find out a bit more. For him [Tarkowski] it's an everyday thing."
Jackson added: "Fingers crossed [he's not out for the season]. That is what we want and hope. It is a day by day thing with injuries, some lads respond quicker and heal quicker than others.
"We have a report from the doctor in the morning and they will give us an update and we will speak about it day by day.
"Tarky has been out doing some running today, it is about making the right decision for him and the rest of the group and what we think will be right on the day."
Striker Jay Rodriguez also suffered an issue with his hamstring in the warm-up ahead of last month's comeback victory over Watford at Vicarage Road.
The England cap was absent for the loss at the hands of Steven Gerrard's Villans, but has been out on the grass running alongside club captain Ben Mee.
Sunday's game in the capital, however, is also likely to come too soon for the pair. Jackson continued: "I've seen it before — and the doctor has said to us — you can have scans and they can say it's this, this and this, but it's the symptoms that you're treating more than anything else and the guy that it is.
"I've seen it with hamstrings before where they've said 'he's out, he's done', then a week to 10 days later there's no problem. We just judge it on the person, his symptoms and how he's responding to his treatment."
There is good news on midfielder Jack Cork, though, who could play a part against Antonio Conte's men after recovering from a minor problem in his back.
"We are going to be waiting on a couple," said Jackson. "Jack Cork has trained today, he just had a little thing with his back. It was a little thing, which he just rested for a couple of days. He was back in training today [Friday]."