The Bees prioritised the capture of Kelleher after accepting defeat in their bid to keep Mark Flekken, who is set to join Bundesliga side Bayer Leverkusen.
Brentford will pay an initial fee of around £12.5m for Keller, who has a year left on his Anfield contract, but the deal could rise to £18m if performance-related requirements are met.
The Ireland keeper played over a quarter of the champions’ Premier League matches this season when Liverpool's first-choice Alisson Becker was injured.
His move away from the Merseyside club has been heavily reported, with Giorgi Mamardashvili joining the club in July after Liverpool confirmed the signature of the Valencia keeper last summer.
Kelleher has been of interest to a number of Premier League clubs, with [Newcastle](https://www.thenorthernecho.co.uk/sport/football/newcastleunited/) having been linked and Leeds United reportedly keen.
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Leeds will prioritise the signing of a new goalkeeper this summer after winning promotion to the Premier League, with error-prone Ilan Meslier paying the price for some costly mistakes by losing his place in the side to former Newcastle keeper Karl Darlow late in the campaign.
Darlow started the last seven games of the Championship season but Leeds are looking to sign a new No.1 this summer and are now said to have set eyes on Newcastle's Nick Pope.
Pope has been Newcastle's No.1 for the last three years but the Magpies will make a renewed push to sign Burnley's James Trafford this summer.
Trafford was a key target 12 months ago and Newcastle were close to landing the keeper for £20m before the deal fell through.
Burnley's valuation of Trafford is now said to have risen by at least £10m and Newcastle aren't the only admirers of the keeper. But the former Manchester City youngster was keen on the move to Tyneside last summer and the Magpies can offer Champions League football.
What's not clear is what a successful pursuit of Trafford would mean for Pope.
Martin Dubravka signed a new contract in February on the back of his successful stint in the side in the absence of Pope.
Newcastle also have Odysseas Vlachodimos on their books but the 31-year-old only played 45 minutes of football last season after his arrival as part of the deal that saw Elliot Anderson sign for Nottingham Forest, and he doesn't feature in Howe's plans.
Newcastle were heavily linked with Joan Garcia last week but the Espanyol keeper is set to sign for Barcelona, which was always expected to be the case.
Trafford, who is currently away with the England squad, was quizzed on his future at the end of his hugely impressive season with Burnley, saying: "I've got an incredibly good circle around me. It's very small but it's incredibly good. When the season ends I'll go on holiday and relax from football and then I'll start getting ready for the national team in the summer.
"I don't go on my phone too much. I'm not naive enough to say I don't see any of the noise around me but it doesn't bother me. Whatever happens will happen. I've done all I can this season."