The insurance company Direct Line has accepted an improved offer from its bigger UK rival Aviva, in a deal valuing the business at £3.6bn.

Aviva, the UK’s largest insurer, has succeeded in reaching a preliminary agreement to take over Direct Line after submitting a third cash and shares bid valuing London-listed Direct Line at 275p a share.

Aviva had its first offer, at 250p, rejected last week, and this week raised it to 261p a share.

Direct Line’s board, led by the chair, Danuta Gray, rejected Aviva’s first approach last month, saying it was “highly opportunistic” and “substantially” undervalued the business.

In a joint statement on Friday, the board of Direct Line said it remained “confident” in its prospects as a standalone company and continued “to have the conviction and capabilities of the newly established leadership team to deliver the announced strategy”.

However, the board added that the offer price meant that it was minded to recommend to shareholders that they accept a formal offer, and that the combined company would provide the “opportunity to deliver significant synergies creating substantial additional value for both sets of shareholders”.

Dan Coatsworth, an investment analyst at AJ Bell, said Aviva had played the negotiations well, starting with a lower offer it knew would be rejected.

“Aviva had no choice but to dig deeper if it wanted to secure Direct Line,” he said. “The next test is to see if shareholders push for more. Aviva has performed every step of the takeover dance flawlessly. It’s spotted a rival going through a weak phase and thrown its hat into the ring as an interested buyer with a low-ball price to test the water. Now come back with a higher and fairer offer.”

A takeover by Aviva, would create a group dominating more than a fifth of the motor insurance market and 15% of the home sector. The deal would give Aviva a stronger position in car insurance, where Direct Line is the No 2 behind Admiral.

Under City takeover rules, Aviva has until 5pm on Christmas Day to table a firm offer for Direct Line or walk away.

The deal is likely to draw scrutiny from the competition watchdog and insurance supervisors at the Bank of England.

The Kent-based Direct line said earlier this month that it planned to cut 550 jobs as part of a turnaround plan aimed at saving £50m next year. It lost almost 400,000 car insurance customers in the past year.

Adam Winslow joined as the chief executive in March from Aviva, where he ran the general insurance business in the UK and Ireland.

Winslow claimed last weekend that his new management team was “making excellent progress in the early stages of a significant turnaround”. He argued that the company, which owns Churchill, Green Flag and Darwin, had “great brands” and strong prospects on its own.

Direct Line’s co-founder Sir Peter Wood said this month that the company had lost its way and a turnaround would take three to four years, arguing it should accept an offer if Aviva raised it to 275p a share – the value of Friday’s recommended offer.

Direct Line shares rose 7% to 253p on Friday, valuing the company at £3.3bn.

In February and March, Direct Line rejected two takeover approaches from the Belgian insurer Ageas, the second valuing it at £3.2bn, or 239p a share, saying it was “uncertain and unattractive” for shareholders.

The UK insurer was spun out of Royal Bank of Scotland in 2012 and listed on the London Stock Exchange two years later.

Aviva has sold businesses in France, Italy, Asia and elsewhere to focus on the UK, Ireland and Canada under the chief executive, Amanda Blanc. In March, it announced it was returning to the Lloyd’s insurance market through the £242m purchase of Probitas and completed the deal in July.