Tavernier was forced off with a hamstring issue in Boro’s defeat at Millwall last month while Fry picked up an ankle injury in training.
Boro will also be without midfielder Sam Morsy and defender Anfernee Dijksteel when they travel to Oakwell, but Warnock says the latter is making good progress following an operation on his ankle.
When asked about injuries, Warnock told the Mail: “I think Dael will come back before Tav, I think Dael is going to be a couple of weeks.
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“Tav is probably around three weeks, we might get him back for the last two games but I don’t know, roughly that sort of time.
“Dijksteel looked good yesterday, everyone in the club is watching his weight and he looked quite slim yesterday so we were quite pleased about that.
“We have to keep an eye on him the next few weeks because he’s got to come out firing on all cylinders for next season.
"All will be back for pre-season, touch wood.”
The Boro boss added: “The players have got six games to make their case to me - I think one or two players have got to show me desire.
"I think one or two players I’ve been disappointed with stats, I’m not a stats man but I need to see a lot more effort from certain players next season so that’s what I’ll be looking at in these next six games.”