The pair both finished the season with Blackpool, but they are due to be out of contract this summer.
Both players have extensive Championship experience, something which Brentford boss Mark Warburton is keen to introduce to his squad.
Keogh was on loan at Blackpool from Millwall, but it is unlikely the 27-year-old will sign a new deal at The Den and a switch across London for the Ireland international striker is now on the cards.
And Warbuton is also eyeing a deal for 25-year-old Basham, who has been with Blackpool since 2010.
The defender or midfield offers excellent versatility and it is believed that he and Keogh are set to be among Warbuton's main targets.