LONDON, Dec 1 (Reuters) - NatWest has offered the bulk of its 41,500 staff in Britain a minimum 2,000 pound salary increase and a one-off payment of 1,000 pounds, to help them cope with soaring price rises, a source with knowledge of the matter told Reuters.

Companies across Britain are grappling with how much to increase salaries as the end of the year nears and when inflation is at a 41-year high of 11.1%, squeezing workers' budgets.

The pay offer would apply to 39,000 staff on lower pay bands in Britain and comes after the bank struck an agreement with the Unite union and the Financial Services Union. The terms are dependent on a ballot of their members.

"We are committed to paying our colleagues fairly and providing further support during a difficult time," NatWest CEO Alison Rose said in a statement.

NatWest's rivals Barclays and HSBC are separately facing union demands for above-inflation pay rises for their staff in Britain, according to a separate source with knowledge of those discussions, adding talks were ongoing.

Barclays and HSBC declined to comment.

Britain's biggest domestic bank Lloyds offered staff a minimum 2,000 pound pay rise earlier this month. A Lloyds spokesperson said the deal had since been approved for 2023.

The prospect of higher bank wage bills has put investors on edge. NatWest's shares plunged as much as 10% when it warned of rising costs next year in its third-quarter results on Oct 28, which CEO Rose put down to the need to pay staff more.

Banking analysts at RBC removed their outperform rating from NatWest's stock last week partly due to cost concerns, and predicted NatWest's costs could rise 7% to 7.1 billion pounds ($8.7 billion) in 2023. Staff wages make up around half of banks' costs, the RBC analysts said. (Reporting by Iain Withers; Editing by Elaine Hardcastle)