Randall toured New Zealand with England this summer and is hoping to feature in the November internationals.
Current England number nine Alex Mitchell is battling to overcome a neck problem in time for the opener against the All Blacks on November 2.
And England head coach Steve Borthwick will consider the claims of players like Randall, who made his Test debut in 2021, Bath scrum-half Ben Spencer and Leicester's Jack van Poortvliet.
Randall gave a reminder of his qualities with an influential display against Bath at the Recreation Ground.
"I have a huge demand on our nines, they have to be the fastest on the park," Bristol rugby director Lam said.
"One on the feedbacks from Steve was he wanted the nines to be quicker.
"100 per cent, Ben Spencer is a really good player, but his teams tend to kick more. Harry's strength is his speed, his sniping and his ability to put teams under pressure.
"If they want a like for like for Mitchell, if he is out, then Harry's that man."
Bath's first defeat of the season came after a dominant first-half display from a Bristol side inspired by flanker Santiago Grondona.
He scored two tries, while there were also touchdowns for Max Malins, Benhard Janse van Rensburg and Gabriel Oghre, with AJ MacGinty kicking four conversions and a penalty.
Click here for all of our details on the match.
Elsewhere, Leicester scored seven tries to rout struggling Newcastle 42-10 at Kingston Park.
The Tigers' second win this term condemned bottom of the table Newcastle to a 24th consecutive defeat in the Premiership.
Leicester were without head coach Michael Cheika due to a one-match ban but his absence didn't stop them running riot thanks to tries from Ollie Hassell-Collins, Anthony Watson, Emeka Ilione, Freddie Steward, George Martin, Will Wand and Hanro Liebenberg.
