Mr Bourne said he had 'made his peace' with Mr Morgan
Discussions will take place over the weekend concerning the position of Welsh Conservative leader Nick Bourne, the BBC has learned.
It follows the row over a party dossier attacking Rhodri Morgan, from which Mr Bourne has dissociated himself, after it was said to be signed off by him.
The document called the first minister the "Clown Prince of Wales" and criticised his dress sense and hair.
But Mr Bourne later said it went too far and it had not been issued by him.
BBC Wales Welsh Affairs Editor Vaughan Roderick said although there was some backing for Mr Bourne's position from assembly members, informal discussions will take place at a senior level in the party this weekend ahead of a meeting of its assembly group on Tuesday.
Earlier Mr Bourne told the Richard Evans programme on BBC Radio Wales: "We have good personal relations and I hope we always will.
"It was a dossier that was issued by my party and it was not issued by me personally.
I did think in certain respects it had gone too far
Nick Bourne, Welsh Conservative leader
"I spoke to Rhodri as soon as I got back on Tuesday (from the Conservative conference) and sought him out and we had a very good conversation.
"I did think in certain respects it had gone too far. I told him that.
"We have our political differences and let's focus on those.
"I have given a very clear instruction to my assembly members and to my staff that is what we do."
The 39-page document claimed Mr Morgan's "behaviour and comments" had become "increasingly erratic and unusual."
It said he was "noted for his unique dress sense and hairstyle".
The dossier also listed occasions when it suggests "eyebrows were raised by his attire."
For example, the document observed that Mr Morgan wore a "woolly red jumper and beige cargo pants" at a function in honour of the former Swansea West Labour MP Donald [now Lord] Anderson in 2005.
The dossier accused Mr Morgan of looking to the past rather than preparing for the future, said he had blamed others for his "own shortcomings" and been guilty of "storing up trouble for the future, for his successors, and for Wales."
However, Conservative sources have contradicted Mr Bourne's version of events.
They claimed the dossier, which was issued in Mr Bourne's name, had been "signed off" by him.
Speaking shortly after the document was issued last weekend, Mr Morgan said he had heard that "there is some really unpleasant political stuff going on and you just have to accept that."
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.