The Commons leader said he did not want to be "prescriptive" but he believed that covering people's faces could make community relations more difficult.
Mr Straw has said he asks Muslim women at his Blackburn constituency surgeries if they would mind removing veils.
Some Muslim women called his remarks insulting, but other Muslims said they understood his concerns.
Mr Straw has dismissed suggestions that his remarks are designed to raise his profile ahead of Labour's deputy leadership election.
He has yet to confirm whether he will join the race to succeed John Prescott but is widely expected to do so.
Meeting strangers
Mr Straw is Labour MP for Blackburn, where between 25% and 30% of residents are Muslim.
He sparked controversy when he told his local paper he asked female constituents visiting his surgery if they would uncover their faces.
Find out about different styles of Muslim veils and headscarves
In graphics
He said he made sure he had a female colleague in the room when asking someone to show their mouth and nose - and his constituents had so far always agreed to do so.
Asked on BBC Radio 4's Today programme if he would rather the veils be discarded completely, Mr Straw replied: "Yes. It needs to be made clear I am not talking about being prescriptive but with all the caveats, yes, I would rather."
Mr Straw explained the impact he thought veils could have in a society where watching facial expressions was important for contact between different people.
"Communities are bound together partly by informal chance relations between strangers - people being able to acknowledge each other in the street or being able pass the time of day," he said.
"That's made more difficult if people are wearing a veil. That's just a fact of life.
"I understand the concerns but I hope, however, there can be a mature debate about this.
"I come to this out of a profound commitment to equal rights for Muslim communities and an equal concern about adverse development about parallel communities."
'Separateness' fears
Mr Straw stressed it was a choice for women and he was making a request and not a demand.
"What I've been struck by when I've been talking to some of the ladies concerned is that they had not, I think, been fully aware of the potential in terms of community relations," he said.
HAVE YOUR SAY Of course Jack Straw has a valid point... Communication isn't just verbal it is also to do with body language and facial expression
Dave, Reading
Send us your comments Why women wear veils
"I mean, they'd thought of it just as a statement for themselves, in some cases they regard themselves as very religious - and I respect that - but as I say, I just wanted to put this issue on the table."
He said he was worried the "implications of separateness" and the development of "parallel communities".
Tony Blair's official spokesman said the prime minister "believes that it is right that people should be able to have a discussion and express their personal views on issues such as this".
The spokesman said Mr Straw's comments were not government policy and he refused to reveal Mr Blair's views on the issue.
'Dangerous doctrine'
Mr Straw's comments have provoked a mixed response from Muslim groups.
The experience of visiting their MP is difficult enough for many people without having to consider a dress code
Simon Hughes,
Liberal Democrat president
Reaction to Straw remarks Veil torn from Muslim woman
The Islamic Human Rights Commission called Mr Straw's views "astonishing" and accused him of discrimination.
The Protect-Hijab organisation said the "appalling" comments showed "a deep lack of understanding".
But Dr Daud Abdullah, of the Muslim Council of Britain, said he could understand Mr Straw's discomfort adding that women could choose to remove the veil.
Labour's Baroness Uddin said debate was needed but perhaps not in the way Mr Straw had framed it.
She was worried he had talked about veils being a statement of separation - that acknowledged the government's failure to ensure Muslim women were "part and parcel" of British society, she argued.
Conservative policy director Oliver Letwin said it would be "dangerous doctrine" to tell people how to dress.
And Liberal Democrat president Simon Hughes: "The experience of visiting their MP is difficult enough for many people without having to consider a dress code."
niqab.
The Rt. Hon. Jack Straw, Leader of the British House of Commons, has urged Muslim women not to wear the full veil or Niqab. Initially, he said that, when meeting his constituents, he found it difficult to talk to anyone wearing a full veil because the expression on the face conveyed much of what a speaker meant. Later, he expanded his opinion saying that the wearing of the veil at any time tended to promote separation of peoples and the "formation of parallel communities". 

A 10 Downing Street statement said that this view was Straw's personal opinion and not an expression of the government's policy.

Straw's remarks have elicited a range of responses from members of the Muslim community in the UK. While Daud Abdullah, from the Muslim Council of Britain, said that the choice of wearing a veil was up to individual women, Muslim Association of Britain said that Straw and the Conservative leader David Cameron, who expressed a similar view, were pandering to the "right-wing media". The British public appear to support his view with a number of media polls showing a vast majority in agreement.  A telephone poll by the BBC also showed overwhelming public sympathy, with 93 per cent supporting his views.