No transfer fee was given but Burnley said Brady cost more than their previous record outlay for his compatriot Jeff Hendrick last August. British media reported a trasnfer figure of 13 million pounds.
Brady, who started 34 Premier League games for Norwich last season before they were relegated to the Championship, had a fine Euro 2016, scoring a late winner against Italy that sent Ireland through to the knockout stages.
Schooled as a winger at Manchester United, the midfielder can also play at left back.
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Westwood is also returning to the top flight after joining the Lancashire club from second-tier Aston Villa for a fee of about five million pounds according to Sky Sports.
Burnley said the 26-year-old, who had four seasons with Villa before their relegation last year, also signed a 3-1/2 year contract with the option of another year.
"The manager at Villa (Steve Bruce) said I could speak to Burnley and for me that was perfect -- a fresh start because things had got a bit stale at Villa," Westwood told the Burnley website (www.burnleyfootballclub.com)
"I have had some great times at Villa but recently it's not been too good. I'm back in the Premier League and I've got another chance," added the midfielder, who will be eligible for Saturday's league game at Watford.
(Reporting by Alan Baldwin, editing by Martyn Herman and Ken Ferris)
