He would appear to have become especially vulnerable after the publication of Assembly Members' expense claims this week, which showed that he had spent more than £5,000 of public money on his bathroom, as well as buying such items as a trouser press and an iPod on expenses.
Whilst it should be noted that such expenditure is entirely within the rules, BBC Welsh affairs editor Vaughan Roderick wrote yesterday that "more than half" of the 12-strong Tory group at Cardiff Bay now believe that he should stand down.
He pointed out that Mr Bourne has been under some pressure since the autumn after the publication of a dossier - which he had approved - making personal attacks on Wales's First Minister, Rhodri Morgan.
Today, a report in Wales on Sunday states that Cardiff North Assembly Member Jonathan Morgan - widely viewed to be the likely successor to Mr Bourne - is refusing to rule himself out of the running to take over, albeit not until there is a vacancy. He said:
“The group has been very supportive of Nick and I’m assuming that is still the case. I wouldn’t be a politician if I wasn’t ambitious and of course I would consider it, but at this moment there is no vacancy.”