At tea on the final day, Yorkshire are 169-3 in their second innings, a deficit of seven, with Hampshire requiring seven wickets for victory during the final session.
The weather has also taken a turn for the worse, with light rain falling at North Marine Road.
Resuming on 27-0 after lunch, Yorkshire made positive progress during the afternoon.
Adam Lyth and Jacques Rudolph took their opening stand to 87 before the former fell in the 18th over, caught at short-leg for 44 off a short ball by Dominic Cork.
Rudolph went to a half-century from 74 balls with nine fours before slapping David Balcombe to Michael Carberry at point for 54.
Yorkshire slipped to 147-3 in the 30th over when Anthony McGrath was caught at first slip off the left-arm spinner Danny Briggs for 20.
At the break, Andrew Gale is unbeaten on 15 and Jonathan Bairstow undefeated on 17.
The visitors declared 30 minutes before lunch on the final day on 498-6, a lead of 176.
Hampshire have given themselves a minimum of 71 overs to try to bowl out Yorkshire and, at the break, the home team are 27-0 from six overs.
Earlier, Jimmy Adams and James Vince narrowly missed out on double centuries as Hampshire accelerated after resuming on 367-3.
Adams fell for 196 when he was bowled by Steve Patterson, ending a Hampshire record fourth-wicket partnership of 278 with Vince, eclipsing the 263 recorded by Roy Marshall and Danny Livingstone against Middlesex at Lord's in 1970.
It was also the highest fourth-wicket stand in first-class matches at Scarborough, beating the 258 by Yorkshire's Phil Robinson and David Byas against Kent in 1989.
Vince perished for 180 when he pulled a rank long-hop from Adil Rashid to Jonathan Bairstow at deep mid-wicket after adding 63 in eight overs with Sean Ervine.
When Ervine was run-out by Patterson for 30, Hampshire captain Dominic Cork called a halt.