Straw, whose Blackburn, northwest England constituency has a 22-percent ethnic minority population, said he had expected his request would cause problems, but has found that most women are "relieved" to comply.
"I defend absolutely the right of any women to wear a headscarf," Straw said in comments that appeared in his regular local newspaper column in the Lancashire Evening Telegraph.
"As for the full veil, wearing it breaks no laws.
"I go on to say that I think, however, that the conversation would be of greater value if the lady took the covering from her face.
"Indeed, the value of a meeting, as opposed to a letter or phone call, is so that you can – almost literally – see what the other person means, and not just hear what they say."
He said he now makes sure he has a female member of staff with him during meetings in his office.
"I can't recall a single occasion when the lady concerned had refused to lift her veil; and most I ask seem relieved to have done so."
However, Muslim community leaders were angered by the remarks.
'Shocked and dismayed'
Massoud Shadjareh, chairman of the Islamic Human Rights Commission, was astonished that Straw does not realize his job "is to represent the interests of the constituency, not to selectively discriminate on the basis of religion."
Nasrullah Anwar, spokesman for the Council of Mosques, told Sky News that "I was shocked and dismayed by his comments.
"You would think that someone in his position would be a little more sensitive and more understanding and perhaps better advised on what is an acceptable means of communicating your wishes and what's not," Anwar said.
Anwar said the issue revolved around respecting the wishes of the woman who has chosen to cover her face, adding that some Muslims considered wearing the full-face veil as obligatory.
Straw was moved from being foreign secretary to the government business management job of leader of the House of Commons in May, widely seen as a demotion.
After Prime Minister Tony Blair, he is the politician most closely associated with Britain's decision to join the invasion of Iraq in 2003.
In March, when he was still foreign secretary, Straw took United States Secretary of State Condoleezza Rice to Blackburn, where the pair were greeted by a series of anti-Iraq war demonstrations.
He acknowledged in his article that his comments could provoke controversy.
"I thought a lot before raising this matter a year ago," he said.
Two years ago, Muslim schoolgirl Shabina Begum lost a High Court battle for the right to wear a long hijab, which does not include a veil, to school in Luton, central England.