The Surrey slow-left-armer, 24, was seen being congratulated by team-mates during training on Wednesday.
He could replace off-spinner Gareth Batty, who took 4-116 in England's 22-run win in the first Test.
Seamer Stuart Broad may be rested, with Steven Finn or Jake Ball likely to replace him.
England play five Tests in India after the Bangladesh series, and England captain Alastair Cook has said it is likely that the squad will be rotated this winter.
Ansari missed the chance to make his England debut in 2015 after dislocating his left thumb, and injured the thumb again in May this year.
"The pace that he bowls and the control that he has can only be a positive thing for us," England wicketkeeper Bairstow told BBC Sport.
"The bite that he'll get out of the wicket being a left-arm orthodox bowler - it's a good asset for us."
Bairstow broke the record for the most runs scored by a wicketkeeper in a calendar year during the first Test, surpassing Zimbabwean Andy Flower's tally of 1,045 in 2000.
Bairstow, who has scored 1,091 runs this year - England are set to play six more Tests - said: "It's not sunk in yet. It's something that won't sink in until after the year's gone out.
"There's so much cricket still to be played this year. But I'm absolutely delighted to have scored so many runs this year and hopefully it can continue."
Bairstow, who has been batting at seven, praised the depth of the England squad, saying he is just "happy to be in the side".
He added: "It's great to be surrounded by such great players.
"If you look at myself, [Chris] Woakes, [Adil] Rashid - within those three guys you've got something like 40 first-class hundreds coming in at seven, eight and nine.
"That's a great place to be, with the strength and depth we're showing with our batting line-up."
Spinner Zafar Ansari is set to make his England debut in the second and final Test against Bangladesh in Dhaka starting on Friday.