Cook's latest opening partner is the most pressing matter to resolve with 19-year-old Haseeb Hameed and Ben Duckett, 22, in contention to get the job in place of Alex Hales, who opted to miss the tour due to security concerns.
Hameed is favourite having enjoyed a prolific county season with Lancashire in which he batted with great maturity and displayed a sound defensive technique.
"There's a really nice calm aura about him when he's practising, and certainly in the middle, when I've played against him twice this year," England fast bowler Stuart Broad said.
"He knows his gameplan for such a young guy and has been hugely impressive in the conditions so far."
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Duckett is a more attacking player, a fierce cutter and puller of the ball, and many believe he would provide a more suitable foil for the obdurate and disciplined Cook, particularly against the inexperienced Bangladesh seam bowlers.
England could opt to play Duckett in the middle order in place of Gary Ballance, who has not cemented his place after being recalled to the side this year.
"Duckett scores in slightly different areas, he's quite awkward to bowl to and he sticks to his gameplan," Broad said.
Bangladesh, who are playing their first test in nearly 15 months, have improved significantly in limited-overs cricket, especially at home where they did not lose a bilateral ODI series in two years until England snapped that streak last week.
They have traditionally struggled to sustain the intensity over five days, however, and the lack of experience in their pace attack also compromises their ability to take 20 wickets in a test.
The hosts have packed four specialist spinners into their squad and skipper Mushfiqur Rahim will be counting on all-rounder Shakib Al Hasan to help the side prove they belong at test level.
The first of the two tests starts on Thursday in Chittagong.
(Reporting by Ed Osmond in London and Amlan Chakraborty in New Delhi; editing by Peter Rutherford)
