Seventeen student bodies have endorsed the BDS movement - which calls for an international boycott of Israel over the way it treats Palestinians.
Some Jewish students in the UK say growing support for BDS has fuelled a rise in anti-Semitism on campuses.
The Commission said it would assess the concerns and take action if necessary.
The BDS - which stands for Boycott, Divestment and Sanctions - describes itself as a human rights organisation and criticises Israel for its human rights record.
It says it stands for "freedom, justice and equality", saying it is "inclusive and categorically opposes as a matter of principle all forms of racism" - including anti-Semitism.
Devora Khafi, a student at Queen Mary University of London, is one of nearly 10,000 Jewish students in the UK.
She told the BBC's Victoria Derbyshire programme she felt physically intimidated when other students hijacked a speech at London's UCL given by a former Israeli defence force officer.
She said she was pushed against some doors, jostled and shouted at by protesters.
She says she fears having a connection with Israel has made other people at her university hostile towards her.
"If I ever express a viewpoint in class or on social media, I get comments, in public or private," she said.
"It has been hard. It's been two years of constant fighting for our freedom of speech, our rights, our wellbeing. I've missed deadlines, I've had counselling, I've had anxiety episodes," she said.
Devora believes the BDS movement is adding to a climate of hostility against Israel on campus, which she says has turned her and other Jewish students into scapegoats.
"If you look behind the BDS lens, it calls for these things that are not peaceful. They harm Jewish students and pro-Israel students and they don't support a peaceful, atmosphere on campus."
Support for BDS has been growing rapidly on British campuses.
London's School of Oriental and African Studies (SOAS) students' union in London was one of the first to endorse the BDS, after students voted in favour in a referendum in 2015.
The students' union implemented an academic boycott of Israel.
The BBC found 17 students' unions have now passed motions to endorse the BDS boycott of Israel.
They include Belfast, Birkbeck, Brunel, Essex, Exeter, Goldsmiths, Kings College London, Kingston, Lancashire, Liverpool, Manchester, SOAS, Strathclyde, Sussex, Swansea, University of Arts London and UCL.
But it is possible they may be breaking the law by backing the movement.
Since 2010, students' unions have become registered charities, meaning they have a legal duty to represent the interests of their members.
According to guidance, public comment should also be limited to matters that affect the welfare of the union's members as students.
The BBC understands these issues have been raised with the Charity Commission, which is assessing them to determine if any regulatory action should be taken.
A Charity Commission spokesperson said concerns about links to the BDS Movement would be "assessed consistently" and, where appropriate, it would "take regulatory action".
David Holdsworth, from the commission, said students' unions must not be discriminatory towards students "of a particular faith or race".
Isaac Bentata, a student at UCL, said there were times when he didn't feel comfortable expressing his views as a Jewish student.
"During my first week at university all I saw were BDS stickers", he explained.
He said he was accused of being a "Mossad agent" after speaking up at one BDS event.
"As we left the event we saw that we were being followed by this guy and we had to run to the station because we'd heard about him being particularly hostile," he added.
"I thought, 'this is not how I'm supposed to be feeling because I asked questions during a debate at a university'."
Both the NUS and the BDS National Committee declined to comment to the BBC.
Of the students' unions we contacted, only SOAS provided us with a response.
Its co-president, Ayesha Abbasi, said it was a "democratic organisation" and followed "democratic decision-making processes."
"We have had no formal complaints of anti-Semitism or bullying from any Jewish students since the BDS referendum," she added.
Sai Englert, a Jewish PHD student at SOAS and a member of the university's Palestine Society, said: "The idea that somehow supporting BDS, supporting boycott etc is a blanket boycott on individuals I think is very dangerous. It's not.
"It's about saying 'we don't want institutional links, economic links, political links with institutions, governments, companies that are complicit in attacks on rights'."
Watch the Victoria Derbyshire programme on weekdays between 09:00 and 11:00 on BBC Two and the BBC News channel.
Concerns over some students' unions' support for a boycott of Israel are being looked into by the Charity Commission, the BBC has learned.