Oxfam has suspended all operations with a call centre following allegations that it uses high-pressure fundraising techniques.
The Fundraising Standards Board (FRSB) said it is "deeply concerned" about the claims made and said they will investigate.
Other charities also say they will look into the issues raised in a Mail on Sunday investigation.
Tim Hunter, Oxfam's director of fundraising, said: "We take the allegations contained in the article very seriously and have suspended all operations with both Listen Ltd and Street Academy, pending a thorough investigation.
"Oxfam carries out regular quality checks of fundraising calls on our behalf and insists on high standards of training and monitoring."
Listen says it is an established fundraiser, while Street Academy aims to raise the standards of street fundraising and is part of housing charity Shelter.
In a statement, Listen told the Mail on Sunday: "Listen, an established and multi-award winning fundraiser, which raised over £40 million for UK charities last year, provides training to all employees in accordance with the requirements of the sector's regulators and is looking into the claims, having asked The Mail on Sunday to provide the necessary evidence to assist in this process."
Campbell Robb, Shelter's chief executive, said: "Street fundraising remains an important income stream for the charity sector. Shelter relies on public support and funding for the majority of our activity, and without this we simply couldn't continue to be there for everyone who needs us.
"We take any accusation of bad practise amongst street fundraisers employed by the Street Academy seriously, and undertake regular mystery shopping to make sure that they adhere to our strict code of practice and that of the Institute of Fundraising at all times.
"We will fully investigate this incident and take necessary action to make sure that we continue to maintain our high standards.
"All of Shelter's fundraising activity follows the Fundraising Standards Board's code of practice and we are continually training and monitoring our staff.
"Shelter has a policy to never sell or share our fundraising data, and we receive no data from the Street Academy that isn't associated with our own fundraising activity."
Nick Georgiadis, head of direct giving at Cancer Research UK, said the charity was "deeply disappointed" to hear the allegations.
He added: "Cancer Research UK relies on people's generosity to fund our life-saving work. Our research has fuelled the progress that has seen survival rates for cancer in the UK double in the last 40 years. In the 1970s just a quarter of people survived. Today that figure is half.
"At Cancer Research UK, we are grateful for every donation we receive and would never want anyone who gives us their support to feel pressured in any way.
"We're deeply disappointed to hear these allegations against Listen Limited and we will be investigating this matter urgently."
A Unicef UK spokesman said they expect "high standards" from the agencies who raise money for them and said they are investigating.
He said: "Unicef UK relies entirely on voluntary donations for our work to help children around the world and are extremely grateful to all our supporters.
"We expect high standards from agencies that we employ to fundraise on our behalf. We take these allegations very seriously and are investigating the issues raised by the Mail on Sunday."
The FRSB said: "We are deeply concerned about the allegations that the Mail on Sunday has made. It is critical that fundraisers treat the public with respect, openness and honesty at all times.
"The FRSB will investigate these claims to establish whether any breach of fundraising standards has taken place."
In a statement the RSPCA said: "All of our fundraising activity is conducted in line with the Institute of Fundraising's (IoF) code of practice and we are also an active member of the Fundraising Standards Board (FRSB).
"Both the IoF and FRSB are conducting reviews of current practice within charity fundraising, following concerns raised by members of the public.
"We will be fully involved in these reviews and follow any resulting recommendations to ensure that donors and supporters have confidence and trust in charities and their fundraising."
It comes not long after a children's charity ''wholeheartedly'' apologised after it asked an 80-year-old cancer survivor to donate £100,000.
The Children's Society said it intended to approach ''well-off individuals'' when it launched its fund-raising initiative but a number of people were wrongly contacted.
Retired BT engineer Robert Newman told the Sun on Sunday newspaper that the charity wrote to him twice in seven days, urging him to pay £2,777 a month for three years.
The claims have been made just a few weeks after one of Britain's oldest and longest-serving poppy sellers was found dead after receiving hundreds of letters a month from charities.
Olive Cooke, 92, died from from multiple injuries after jumping into Avon Gorge on May 6. Her family have insisted that - while the letters and phone calls were intrusive and a nuisance - the charities were not to blame for Mrs Cooke's death.