Western Australia were 244 for five at stumps after Tasmania declared its first innings closed at a sporting 384 for five in an effort to set the match up for an outright result on Friday.
That tally was 178-runs behind Western Australia's first innings lead.
Western Australian left-hander Marcus North was an unbeaten 82 as he eyed his second century of the match, with all-rounder Kade Harvey on four.
Western Australia look set to allow North time to post another century on Saturday which means Tasmania will face an even bigger run chase for victory.
Tasmania's best-ever run chase was against Western Australia in Perth in 1995-96 when they succesfully overhauled 402.
Tasmanian captain Dan Marsh said he was hoping Western Australia would declare early on the final day and set a target of about 420 off 96 overs.
"They may bat on for a little bit longer, but they obviously need time to bowl us out," he said.
Tasmania's top wicket-taker was left-arm orthodox spinner Xavier Doherty, who snared former Zimbabwean international Murray Goodwin for 22 and keeper Ryan Campbell for 13 to finish with 2-79 off 16 overs.
Earlier in the day, strapping all-rounder Shane Watson scored his third century of the season for Tasmania off 163 balls with 13 fours and one six and a total of 139.
A hard-hitting Michael Dighton was not far behind, bringing up a century for Tasmania off 178 balls with 20 fours before he fell lbw to North for 101.
Hobart (day three): Tasmania 384-5 decl (S Watson 139, M Dighton 101; B Edmondson 2-54, J Taylor 2-88) v Western Australia 562-6 decl and 244-5 (J Langer 84, M North 82; X Doherty 2-79, J Cox 1-44).
WA puts Tasmania on the back foot