The Glovers currently sit one point outside the top seven following a five-match unbeaten run and take on Colchester United side winless in nine on Saturday (3pm).
A win at Huish Park could see Yeovil jump as high as sixth and Way revealed his players intend to be in a similar position come the end of the season.
He told YTFC’s official YouTube account: “Ben Whitfield put me under massive pressure to say as a group of players they want the play-offs and I have to back them in that department.
“We’ve got that opportunity to get into the play-offs on Saturday so unlucky for Ben, he’s thrown down the gauntlet for the players!
“We have to make sure we live up to that and we’ll be doing everything we can.”
Yeovil return to league action following their Emirates FA Cup first round replay defeat on penalties to Solihull Moors.
Way is keen to move on from the cup shock and insists his side have already achieved a lot in the past year.
“I want the supporters to stick behind this group because you never know what we could achieve,” he said.
“We’ve got 30 games to go, we’re a point off the play-offs, we’re still in the Checkatrade Trophy against MK Dons, and I feel the group have given everything.
“It’s a complete turnaround from last year and want to keep that momentum going. We have to keep believing about where we want to go.”
Fans can purchase tickets for this weekend’s match online or by phone on 01935 847888.