The source said no decisions had been made but a few hundred troops could be sent.
The Ministry of Defence said it was considering "additional support" but also said no decisions had been made.
Meanwhile Defence Secretary Michael Fallon has said Britain will supply ammunition to the Kurdish Peshmerga.
Details of the ammunition have not been announced but BBC defence correspondent Jonathan Beale said it was likely to include bullets for heavy machine guns already sent from the UK. The Peshmerga say they have run out of ammunition for the weapons.
About 300 UK military personnel are currently in the country training Iraqi security forces and Peshmerga fighters.
The Ministry of Defence said the US-led coalition fighting IS had asked Britain to "consider a range of options" to support Iraqi security forces.
It added: "Any additional support will continue to be behind the wire."
At a meeting in Germany on Wednesday, members of the coalition will discuss stepping up the fight against IS.
Britain could send hundreds more troops to Iraq in training roles to boost the fight against so-called Islamic State, a defence source has told the BBC.