The Government has been criticised for not publishing its full report into funding for extremism, which found that Islamist extremist organisations are receiving hundreds of thousands of pounds a year.
The Home Office review into the nature, scale and origin of the funding of Islamist extremist activity in the UK found that for a small number of organisations with extremism concerns, overseas funding is a "significant source of income".
Labour's shadow home secretary Diane Abbott said there was a "strong suspicion" the report was being "suppressed to protect this Government's trade and diplomatic priorities, including in relation to Saudi Arabia".
Green Party co-leader Caroline Lucas also attacked the review for not providing details of which countries funding for extremism comes from.
Home Secretary Amber Rudd said she had decided against publishing the report in full for reasons of national security and because of the volume of personal information it contains.
She said the Government would be directly raising issues of concern with "specific countries as part of our wider international engagement on countering extremism and violent extremism".
The review, commissioned in November 2015 by then prime minister David Cameron, found the most common source of support for Islamist extremist organisations in the UK is from small, anonymous public donations, with the majority most likely coming from UK-based individuals.
"In some cases these organisations receive hundreds of thousands of pounds a year," Ms Rudd said.
"This is the main source of their income."
While overseas funding was a significant source of income for some organisations, it was not so important for the vast majority of extremist groups.
But overseas support has also allowed individuals to study at institutions that teach "deeply conservative forms of Islam", the report found.
Foreign support also provides "highly socially conservative literature and preachers" to the UK's Islamic institutions and some of these individuals have since become of extremist concern.
Ms Rudd said the review "gives us the best picture we have ever had of how extremists operating in the UK sustain their activities".
Other findings include that some Islamic organisations of concern are posing as charities to increase their credibility and to take advantage of Islam's emphasis on charity.
Some of these are deliberately vague about their activities and their charitable status.
The Home Secretary said no single measure will tackle all the issues raised in the review and a comprehensive approach focused particularly on domestic sources of support for all forms of extremism is needed.
The Government will also work with the Charity Commission to address the abuse of charities for terrorist or extremist purposes, which will be introducing a requirement on charities to declare overseas funding sources.
Ms Lucas said: "The Government's refusal to publish this report, and this utterly vague statement, are completely unacceptable.
"The statement gives absolutely no clue as to which countries foreign funding for extremism originates from - leaving the Government open to further allegations of refusing to expose the role of Saudi Arabian money in terrorism in the UK."
Ms Abbott said: "The public has a right to know if any governments, foreign or domestic organisations or individuals are funding extremism in this country, and what the UK government intends to do to prevent that.
"Of course, security intelligence should not be compromised but this is easily achieved by redaction and other means. The Government would never have commissioned this report if it considered this problem insurmountable.
"Instead, there is a strong suspicion this report is being suppressed to protect this Government's trade and diplomatic priorities, including in relation to Saudi Arabia.
"The only way to allay those suspicions is to publish the report in full."
Liberal Democrat leader Tim Farron also said further detail was needed.
"Instead of supporting the perpetrators of these vile ideologies, the Government should be naming and shaming them - including so-called allies like Saudi Arabia and Qatar if need be," he said.
"It seems like the Government, yet again, is putting our so-called friendship with Saudi Arabia above our values. This shoddy decision is the latest in a long line where we have put profit over principle."
Asked about claims that the full report was being kept secret in order to avoid scrutiny of any Saudi involvement, the Prime Minister's official spokesman said: "It is a classified report and there is a large volume of personal information in it and on national security grounds it would not be appropriate to publish.
"The appropriate privy councillors from opposition parties can go to the Home Office and read the report on Privy Council terms.
"The key finding of the report was that the most common source of support for Islamist extremist organisations in this country was small anonymous donations."