Simon Barrett's speaking after the US president's press secretary Sean Spicer accused former president Barack Obama of commissioning GCHQ, Britain's communications monitoring organisation, to spy on Mr Trump.
GCHQ called the claims "nonsense" and "utterly ridiculous". Barack Obama has previously denied spying on Mr Trump.
With its denial, the spying organisation broke a longstanding tradition of not commenting on accusations or stories. It famously said nothing when whistleblower Edward Snowden accused it of having the power to remotely switch on people's mobile phones and use the microphones to listen to them.
Downing Street has since had assurances from the White House that it will not make accusations that GCHQ spied on Donald Trump again.
Speaking on Premier's News Hour, Simon Barrett said: "He's having domestic problems at the moment in trying to enforce his travel ban, so he's wanting to come out and say 'I'm under fire, I'm under attack, this is the Establishment working against me, I can't put my policies in place'.
"So foolishly he's actually bought GCHQ into the equation, which wasn't a very sensible move for our relationship between and the United States on important intelligence gathering."
Listen to Premier's Alex Williams speaking to Simon Barrett on the News Hour:
