Reverend Eleanor Hancock from Holy Trinity Church in Carlisle says some people living in her parish would struggle to pay more council tax and she would rather central government stepped in.
Environment Secretary Liz Truss has praised a trial in Somerset where authorities have been allowed to raise council tax by 1.25% on top of a 2% cap, in order to strength flood defences.
Speaking on Premier Christian Radio's News Hour programme, Reverend Hancock said:
"People that were flooded in our area are some of the most vulnerable people that we have, so their ability to pay is very limited. To have something ...in our general taxation system would be far better.
"People in Carlisle are talking about the need to widen rivers, dredge them, put more bends in them again like there used to be [and] have reserviors or a lake for water to go.
"It's a big environmental issue. It's a big challenge and it can't be made by officers of a council.
"I can't see raising council tax intitially being anything like sufficient [enough] to pay for the work that needs doing, even in Carlisle.
"I, as a taxpayer, would be happy to pay more but it's about actually making sure any money ...is spent in the right way.
"What we're learning is that some of the reasons that Carlisle has flooded again is because our rivers haven't been looked after. There's so much to do."
Some parts of Cumbria have already been flooded four times this winter.
The trial in Somerset has seen the average family across six authorities pay £15.
Liz Truss said: "I think if you look at the structure for the Somerset Rivers Authority that now has the shadow precept so they are raising that funding locally and I think there's also a role for that as well.
Asked whether she wanted to see the model introduced in other areas, she added: "I think the Somerset Rivers Authority is a very good model."
Speaking to the Daily Telegraph, Labour peer Lord Clark of Windermere said: "Flood defences are primarily a national responsibility and the Government shouldn't just pass the buck on to local authorities and in turn to local taxpayers."