Hampshire resumed on 18-1 needing 237 runs to avoid an innings defeat and had reached 173-4 when rain arrived at tea.
Tom Alsop made 53 for the hosts and Michael Carberry was on 37 when play was called off at 16:45 BST.
Seamer Lewis Gregory starred with the ball for Somerset, claiming 3-26, as Hampshire closed 82 runs behind.
With Wednesday's forecast for Southampton looking poor, Somerset may well have to settle for their seventh draw in eight Division One games this season.
A draw for bottom side Hampshire would see them close the gap to Surrey to 14 points with a game in hand.
In the two sessions of play possible, 20-year-old Alsop made his second four-day fifty of the season for the hosts before he edged Peter Trego to Marcus Trescothick at second slip.
Captain Will Smith and Liam Dawson were soon dismissed by Gregory to leave Hampshire 121-4, but Carberry, batting at number four, and Sean Ervine (34 not out) steadied things with a watchful stand of 52.
Somerset were left frustrated by Hampshire's batsmen and the weather on a rain-hit third day of their County Championship game at the Ageas Bowl.