Direct Line shares soared by more than 40% on Thursday after the insurer rebuffed a £3.3bn offer from Aviva, prompting speculation that the company could attract a higher offer.

Aviva’s approach, which was revealed on Wednesday night, is the third rejected by Direct Line this year, with the company having snubbed two previous bids by the Belgian insurer Ageas that culminated in a £3.2bn offer in March.

Despite Aviva’s insistence that the deal would bring attractive returns for both companies’ shareholders, Direct Line said the 250p-a-share bid – made up of cash and Aviva shares – was “highly opportunistic and substantially undervalued the company”.

The board said it was backing Direct Line’s executives, and declined to engage further with Aviva. The move sent Direct Line’s share price to 224p on Thursday, making it one of the biggest risers on the FTSE 250.

Aviva argues a deal would create “attractive returns for both Aviva and Direct Line shareholders”.

Direct Line revealed earlier this month that it proposed to cut 550 jobs as part of a turnaround plan aimed at saving £50m next year. The Kent-based company lost almost 400,000 car insurance customers in the past year. Its chief executive, Adam Winslow, joined the company in March from its suitor Aviva, where he ran the general insurance business in the UK and Ireland.

Analysts at Jefferies said they were not surprised that Aviva’s offer was rejected, given it was a “relatively small uplift” on Ageas’s bid.

The Jefferies analysts Philip Kett and James Pearse said: “Previously, we suggested that the capital and expense synergies available to an acquirer mean that an offer of at least 270p would be more realistic.

“With this in mind, while we agree with Direct Line’s rejection of the offer, we do believe that a higher offer might be forthcoming if the board considered engaging with Aviva.”

Others suggested that Aviva, which is led by the chief executive, Amanda Blanc, could be the most appropriate owner of a company such as Direct Line.

Matt Britzman, a senior equity analyst at Hargreaves Lansdown, said: “Direct Line is playing hard to get, again … [but] there’s a case to be made that Aviva is a better suitor, given it already shares markets with Direct Line in the UK, but it’ll need to up its game – and its offer – if it wants Direct Line to take the proposal seriously.”

However, there are no guarantees of a higher offer from Aviva, and KBW insurance analysts, led by William Hawkins, said it was unlikely to go above 300p a share: “We are always cautious about the bidder’s curse, but we believe Aviva’s approach to Direct Line Group is strategically coherent, could offer considerable synergies, and is currently highly financially attractive.

“The main risk for Aviva is that it seems to be stretching an already below-average solvency ratio, so any further generosity would need to come from shares.”