The large 400-year-old paintings by Flemish artist are currently on display at Arbroath Library.
Liberal Democrat councillor Ben Lawrie said the paintings – plus with five other works with a combined value of £670,000 – should be sold off.
He said: “I’m sure they’re of significant cultural value but I don’t see how this benefits the average Angus resident.
“With money so tight, selling them should be an option on the table. This amount of money could stretch a long way. This is something that should be debated.
“There are a lot of important and difficult decisions that have to be made and everything needs to be looked at.”
The other five works are not on display. Four are in the debating chamber at Town and Country Hall in Forfar while another painting is in storage awaiting restoration.
Last month the council was warned by Audit Scotland it needs to make “unprecedented” changes to achieve a financially sustainable position with an estimated funding gap of £38m over the three years from 2018-19 to 2020-21.
Brueghel’s work is much sought after, with Sotheby’s recording several sales in excess of £4m for smaller pieces of work. The council collection was last valued in 2010.
Arbroath councillor David Fairweather mounted a passionate defence of the paintings, saying he would “strangle” anyone who suggested they are sold.
He said: “If our council in their wisdom was to suggest, even in difficult times, to sell the family treasures, I would be vehemently against it. There are superb works of art.
“As far as selling them, if someone mentioned that to me I would want to strangle them.
“How could anyone possibly want to get rid of these works of art? They could be worth £3m or £4m each.
“I think the sale price would be unbelievable at auction, but they are not for leaving Angus as far as I’m concerned.
“Selling them would only benefit the council in the short term. It’s up to the council to balance their books and selling these paintings is not the answer.”
However, Chloe Westley, campaign manager at the TaxPayers’ Alliance, said the council should “explore every avenue” for raising funds.