Finn, who has taken 120 wickets in 33 Tests at an average of under 30 apiece, is set to replace the injured Ben Stokes.
All-rounder Stokes has been ruled out after damaging his calf in England’s 330-run win in the second Test at Old Trafford that levelled the four-match series at 1-1.
Cook said there was still a slim chance England could play two spinners, by giving a home debut to leg-break bowler Adil Rashid in support of off-spinner Moeen Ali, if there was a marked change in pitch conditions between Tuesday and Wednesday’s toss at Edgbaston.
But with rain falling from grey skies above Edgbaston on Tuesday, Cook indicated that England would likely go in with an attack of four seamers plus Ali.
England also had the option of selecting Nottinghamshire quick Jake Ball, who made his Test debut in Pakistan’s 75-run win in the series opener at Lord’s — Finn’s first wicketless Test.
Finn and Ball made way for James Anderson, England’s all-time leading wicket-taker, and Stokes at Old Trafford.
“Finny is coming in for Ben Stokes, and as we were,” Cook told reporters at Edgbaston on Tuesday.
“We’ve gone for a guy who’s got a proven Test record, who’s taken 120 wickets and has a knack of taking wickets,” said Cook of Finn.
Middlesex paceman Finn took a Test-best six for 79 against Australia at Edgbaston last year.