By Noel Randewich

(Reuters) - U.S. stocks racked up their strongest gain in two months on Monday after Federal Reserve Board Governor Lael Brainard stuck to her dovish stance on interest rates and urged caution about removing monetary stimulus too quickly.

Brainard's speech followed earlier comments by Atlanta Fed Bank President Dennis Lockhart and his Minneapolis counterpart Neel Kashkari in which they suggested there was no urgency to raise benchmark U.S. rates.

"Today's new normal counsels prudence in the removal of policy accommodation," said Brainard, a permanent voting member and the last official scheduled to speak ahead of the Fed's policy-setting meeting on Sept. 20-21.

Traders trimmed their views on the likelihood of a September rate hike to 15 percent, from 24 percent on Friday, and for a December hike to 54.5 percent from 59.2 percent, according to the CME Group's FedWatch tool.

"There's not enough ammunition for a September rate rise. The market wants to put this off as long as possible," said Jake Dollarhide, chief executive officer of Longbow Asset Management in Tulsa.

Fears of a September rate hike had sent the three major U.S. stock indexes tumbling on Friday in their worst decline since Britain's June 23 'Brexit' vote to leave the European Union.

Monday's were the strongest daily gains for the Dow Jones industrial average and S&P 500 since early July but the indexes did not fully recover from Friday's steep drop.

The Dow <.DJI> rallied 1.32 percent to end at 18,325.07 points and the S&P 500 <.SPX> jumped 1.47 percent to 2,159.04.

The Nasdaq Composite <.IXIC> surged 1.68 percent to 5,211.89.

About 7.8 billion shares changed hands on U.S. exchanges, well above the 6.3 billion daily average for the past 20 trading days, according to Thomson Reuters data.

The S&P 500 is trading near 17 times expected earnings, above its 10-year average of 14, according to Thomson Reuters Datastream.

High valuations are a risk, but they do not mean stocks are likely to fall in the short term, said Brad McMillan, chief investment officer for Commonwealth Financial in Waltham, Massachusetts.

"It's like high blood pressure: You're probably going to have a heart attack some day, but it doesn't mean it's going to happen anytime soon," McMillan said.

Apple <AAPL.O> rose 2.24 percent, giving the S&P 500 its largest boost.

Perrigo <PRGO.N> jumped 7.34 percent after activist investor Starboard Value disclosed a stake in the drugmaker and said it must make improvements to revive its stock. The stock was the top percentage gainer on the S&P.

Advancing issues outnumbered declining ones on the NYSE by a 2.61-to-1 ratio; on Nasdaq, a 2.63-to-1 ratio favored advancers.

The S&P 500 posted 2 new 52-week highs and 4 new lows; the Nasdaq Composite recorded 32 new highs and 44 new lows.

(Additional reporting by Yashaswini Swamynathan in Bengaluru and Rodrigo Campos in New York; Editing by James Dalgleish and Nick Zieminski)