The 28-year-old has been placed on a free transfer by the Magpies after his recent criticism of the club on the social networking site.
His outbursts quoted the likes of George Washington and George Orwell to underline his frustration and on Tuesday he was forced to train alone.
But he trained with the reserve team Thursday and Pardew insisted Barton could yet stay with the north-east club.
“I’ve never closed the door on anyone and I’m not closing the door on Joey,” Pardew told a news conference.
“He trained with the development team today and, who knows, he may train with the first-team soon.”
Moments later Barton, again via his Twitter page, offered hope of a reconciliation with a reference to the Victorian-era engineer and bridge-builder Isambard Kingdom Brunel.
“Does anybody have Isambard Kingdom Brunel’s number?” he wrote.
Meanwhile Pardew said of the impasse between the player and the club: “It’s a situation I didn’t envisage being in.
“Joey is a great player and you want great players in your team. It’s very important as a football club that you are all pulling in the same direction — at this moment of time where Joey is not pulling in the same direction.
“That’s not to say that can’t change in a short period of time.
“I’m still reflecting on the situation, as is Joey at this present time, on how best to resolve this issue.
“I think that is important at the moment.”
Pardew said but for the use of Twitter, the row may already have been sorted out. “Maybe if it wasn’t for Twitter and this instant media it may have got resolved on Monday morning with me and Joey in my office.
“The problem with Twitter — we need to get a hold of this.
“We have got nothing from the Premier League on how to deal with this.”
Pardew revealed he’d sought advice from Manchester United manager Sir Alex Ferguson on the issue of players using social networking sites.
“I spoke with Sir Alex Ferguson today and he’s had this problem with Twitter before,” he added.
“You cannot be critical of the club within a personal account. You shouldn’t do that on your own account.
“We’re now following the guide that Sir Alex has said they have done (at Manchester United).
“I’m not against those sites, and I don’t think many managers in the Premier League are, but they have to be careful not to mention the football club.”