* Fed Governor Brainard stands by dovish stance
* Fed policy-setting meeting scheduled for Sept. 20-21
* Apple is top boost on all three major U.S. stock indexes
* Indexes up: Dow 1.07 pct, S&P 1.22 pct, Nasdaq 1.39 pct (Adds comment, updates prices)
By Noel Randewich
Sept 12 (Reuters) - U.S. stocks were heading for their strongest session since July 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.
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.
At 2:11 pm (1811 GMT), the Dow Jones industrial average was up 1.07 percent at 18,279.15 points and the S&P 500 had gained 1.22 percent to 2,153.72.
The Nasdaq Composite added 1.39 percent to 5,197.36.
After Friday's selloff, 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.
Following two days of losses, Apple rose 1.76 percent, giving the S&P 500 its largest boost.
Perrigo rose 7.14 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 2.12-to-1 ratio; on Nasdaq, a 2.02-to-1 ratio favored advancers.
The S&P 500 posted one new 52-week high and four new lows; the Nasdaq Composite recorded 18 new highs and 42 new lows. (Additional reporting by Yashaswini Swamynathan in Bengaluru and Rodrigo Campos in New York; Editing by James Dalgleish)