Currently there is no entitlement to leave - even following a child's death.
The prime minister, who himself suffered the death of his son, has said he is happy to look at the issue.
The poll, conducted for the Change Bereavement Leave campaign, showed 71% of the 1,508 people questioned thought the law should be changed.
While many employers exercise discretion when a close family relative of a worker dies, only 15% of those questioned correctly understood there was no guarantee they could take paid time off.
Glasgow South MP, Tom Harris, introduced a Bill in the House of Commons in September this year to introduce a national guaranteed minimum bereavement leave.
It would amend the Employment Rights Act 1996.
Without government support it stands little chance of success, but David Cameron, speaking at Prime Minister's Questions on 20 November, acknowledged it was "an important issue."
His six-year-old son, Ivan, who suffered from cerebral palsy died in February 2009.
He told Mr Harris: "I am happy to look at that, having suffered that experience myself.
"As a member of parliament, it is possible to take a little bit of time to stand back and come to terms with what has happened, because colleagues and the people who help us are ready to step in and do what they can.
"He has raised an important point; let me look at it and get back to him."
The founder of Change Bereavement Leave, Lucy Herd, began the campaign in 2010 after her 23-month-old son, Jack, drowned in their garden pond.
Her then partner was only allowed by his employer to take three days off work - one for the funeral.
She said that was an "inhumane anomaly" which meant it was vital the law was changed.
"David Cameron acknowledged he was able to take two weeks off after the death of his own son, but sadly not all parents have sympathetic or understanding employers or can afford unpaid time off.
"We would like to see four weeks of paid bereavement leave for parents."
Tom Harris said: "This is an injustice that parliament needs to address. I was pleased that at Prime Minister's Questions on Wednesday, David Cameron responded positively to me and seemed keen to consider a change in the law.
"Working from figures available in the Commons library we have come up with a very conservative estimate that at its highest level the cost of the measure to employers across the UK would be Â£4m a year."
He said bereavement leave would allow parents time to grieve, although he recognised for some a return to work and normality might be the way to cope with a personal tragedy.
His Bill proposed a two-week entitlement with the ability for employers to be flexible beyond that.
In one recent case in Birmingham an NHS employee was disciplined for taking time off to attend their child's funeral.
The poll, based on a sample of 1,508 British adults, was conducted by Survation in September this year.
Data was weighted to the profile of all British adults, by age, sex, region, household income and 2010 vote.
Most people would support a national guaranteed bereavement leave after the death of a family member, according to an opinion poll.