Sussex all-rounder Wright played the last of his 46 one-day internationals at the 2011 World Cup but has since become a regular again in England's Twenty20 team.
England's limited-overs coach Ashley Giles will also have the chance to see a clutch of other international aspirants at close hand - including Michael Carberry, who once again demonstrated his short-format prowess on Wednesday night with a century for Hampshire in their Friends Life Twenty20 quarter-final win over Lancashire.
Others with previous international experience in the squad for the matches on August 20, 22 and 24 are Danny Briggs, Jos Buttler, Boyd Rankin, Ben Stokes and James Taylor.
National selector Geoff Miller said: "We are delighted to reward this Lions team with an opportunity to impress the selectors ahead of a number of forthcoming ODI series.
"Many of this squad have been in or around England, Lions, EPP or PEPP squads before, so it is a great chance for them to push for places for the England ODI games later this summer and also this coming winter.
"Looking further ahead, we are also keen to build a strong platform for the 2015 World Cup, so opportunities such as this play a big part in giving the coaches an opportunity to assess the levels of these players first hand."
The Lions are also due to step in to face Australia next week, in place of Northamptonshire, in a two-day match on August 16 and 17 at Wantage Road - after the Steelbacks qualified for Twenty20 finals day.
The squad will be announced early next week.
England Lions (v Bangladesh A) squad: LJ Wright (Captain), GS Ballance, DR Briggs, JC Buttler (wkt), MA Carberry, CJ Jordan, J Overton, WB Rankin, BA Stokes, JWA Taylor, JM Vince, DJ Willey.
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- Michael Carberry