Trump has always rejected accusations of possible ties with Moscow but the latest claims from security sources suggest that the matter has been on the radar of security services for a prolonged period of time.
It has also been claimed that GCHQ was not alone in sharing information about possible interactions between figures associated with Trump and Russians.
Sources told the Guardian that a number of countries - including Germany and Australia - also shared intelligence on the matter.
However, the information allegedly obtained by GCHQ was reportedly not the result of any targeted operation focusing on Trump or those associated with him but instead came from routine activity directed at Russian spies.
There has been fierce speculation over Trump's potential links with Moscow, but the President has always denied having any business ties with Russia, saying there were "no deals, no loans, no nothing".
But Dearlove, who was Chief of the Secret Intelligence Service between 1999 and 2004, told Prospect magazine: "What lingers for Trump may be what deals - on what terms - he did after the financial crisis of 2008 to borrow Russian money when others in the West apparently would not lend to him."
Trump's relationship with Russia has been intensely scrutinised both before and after he was elected.
But his position on claims of personal ties with the country has always been unequivocal. He wrote on Twitter before his inauguration in January: "Russia has never tried to use leverage over me."
Trump has also dismissed allegations that members of his team had contact with Moscow before he was elected, claims Dearlove described as "unprecedented".
Trump has described investigations into possible links between Russia and his presidential campaign as a political "witch hunt" instigated by his opponents.
The latest claims about British intelligence services sharing information relating to Trump and Russia have the potential to reignite a row which erupted after the White House accused GCHQ of helping Barack Obama spy on Trump in the run-up to the US presidential election.
Sean Spicer, Trump's press secretary, repeated claims made by an analyst on Fox News that the former President used the British listening post to spy on Trump Tower.
The comments prompted a furious and unusually strong response from GCHQ, which labelled the claims "nonsense".
