Councillor Peter Nutting said the increase would be kept at a minimum because the council already looks after 5,000 miles of road.
He was speaking after it was announced that the Department of Transport would be providing £54 million for the £71m relief road, between Battlefield Link Road and Oxon Link Road, which will provide a new, single-carriageway route taking traffic out of the town centre.
It is hoped the work, which is likely to start in 2022, will reduce congestion around Shrewsbury and improve air quality in the town.
Councillor Nutting said it was hoped outside help would make up the difference in cost.
"The deal is we're getting £54m towards a £71m project," he said. "We're hoping to get the difference from Marches LEP and from the Combined Authority, but if not Shropshire Council will provide the difference.
"Shropshire Council will be responsible for the upkeep of the road, and yes there will be perhaps a slight increase in council tax but it will be so minimal because we're already looking after 5,000 miles of roads in Shropshire already."
Councillor Nutting was speaking in front of about 100 constituents at a surgery held by Shrewsbury MP Daniel Kawczynski at Oxon Church Hall on Thursday night.
He said a lot more money would be needed to completely solve the problem.
"We are catching up, but we do 100 and another 50 appear," Councillor Nutting said. "It's like chasing an animal that's going too fast, but we are getting nearer.
"The roads in the north of the county aren't in bad condition compared with other areas. South Shropshire has suffered more than anybody.
"This isn't technical language, this is how an engineer put it to me. The roads in the middle and north of the county are proper roads. In South Shropshire a lot of the rural roads are just goat tracks that have had tarmac put over them over the years and with all the heavy tractors they use they are destroying the roads that exist. What we need to do is spend a lot of money actually giving them a proper undersurface and drainage.
"We are catching up. We have had more Government money, an extra £7.3m in Shropshire. That sounds like a lot of money, but there are 5,000 miles of roads we have to look after and in an ideal world we need an extra £50m."