"Well, there aren't too many performance ones," said Nicholson, who in September won the Burghley Horse Trials for the third year running, setting a record for consecutive wins on the same horse.
"I'd very much like to [return] but I have got to feel like I'm wanted back in there as well," Nicholson said.
The elite squad will be reviewed in June.
Nicholson had yet to respond to the letter from president Chris Hodson, QC, so would not go into detail about the conditions but Hodson confirmed they focus on behaviour.
His omission - which cost him a $55,000 High Performance Sport grant - appears to be punishment for the incident.
The falling out resulted from Nicholson, 53, becoming unhappy about the care of his star horse Nereo during the World Games.
After the games, Pynn and high-performance director, Sarah Harris, resigned. Pynn has reconsidered and made himself available through to Rio. Harris told the Herald she had wanted a change after six years and to reduce the time spent away from her young family. "Who knows, I may come back."
Harris and eventing high performance coach Erik Duvander made a complaint about Nicholson to Equestrian NZ about the altercation. The complaint was withdrawn after Nicholson pulled out of the squad.
He later changed his mind requesting that his withdrawal be withdrawn but this was not accepted by the sport's body.
Nicholson said the vet should have remained with his horse while it received a recovery drip to guard against error or sabotage. He otherwise had no problem with the vet. "He's a very good vet," Nicholson said.
"For sure, I did sort of do my nut at the vet for his treatment of my horse at the World Games. I don't apologise for my behaviour when it comes to the well-being of my horse.
"I was at the World Games and I was in with a live chance. I possibly could have done it a bit smoother but that's the way I operate. And now I'm in all sorts of trouble for it."
Only the horses ridden by Nicholson (ninth) and Jonelle Price (fourth) were live of the New Zealand team at the time. Other team members - Mark Todd, Jock Paget and Tim Price - did not finish.
"It's more like my fall-out with Chris Hodson and Erik Duvander," Nicholson said.
Asked whether he could work with Duvander again Nicholson said, "I can work with anyone if I put my mind to it."
Messages left with Duvander were not returned by deadline.
Harris said she did not believe Nicholson had a point about the vet's care of his horse.
"Absolutely not. I would support the vet and his work at the World's 100 per cent."
Hodson said there was no time limit but a response satisfactory to Equestrian Sport NZ was required for the star to be considered for the squad.
"The guy is a very good rider, that's never been an issue. I'm terribly disappointed that this has happened. I'm hopeful [agreement will be reached]. We can do no more than hope."
Nicholson had been frustrated by Equestrian NZ's handling of the positive returned by Paget's horse Clifton Promise to a banned substance at Burghley in 2013.
"They should have explained the process to Paget and left him to it, as Australia did with Kevin McNab, whose horse failed a test to the same drug at the same event," an industry source said.
Instead the body effectively declared Paget innocent of wrongdoing and worked with the rider to explain the presence of the banned substance. An inquiry accepted the positive was due to a tainted supplement.
Hodson stood by the decision to support Paget which he said was based on circumstances including the unlikelihood of using that particular drug to cheat, as well as Paget's word.
Nicholson was also incensed by a statement from Equestrian NZ to the effect that Paget requested that he be disqualified, thereby clearing the way for the title to go to Nicholson whose horse was second, an empty gesture as the presence of the drug in the horse's system meant it was automatically disqualified.
"The view of some was that [the incident at the World Games] was waiting to happen," an industry source said.
Though Nicholson could be volatile and had blown up at a team official at the 2000 Olympics, a sports administration source said officials sometimes lost sight of their core role which was to help athletes deliver their best performance.
It is understood that Alex Baumann, head of High Performance Sport NZ, wants Nicholson back for the Rio Olympics and is encouraging both parties to settle their differences. Nicholson confirmed Baumann had been in touch.
Baumann's organisation seeks to pick the sports most likely to win medals at the Rio Olympics and fund them accordingly. Eventing is on the second tier of funding and receives $1.8 million annually through to Rio.
The eventing team won bronze at the London Olympics, where Nicholson's fourth place was the best individual result. Nicholson has won three team medals at the Olympics but is still chasing his first individual medal.
Equestrian NZ is undergoing change including a review of its governance structure. It has an interim CEO while it seeks a permanent replacement for Jim Ellis, is interviewing to replace Harris and is seeking new selectors.
Story so far
• August/September: Nicholson had an altercation with the NZ team vet over the care of his horse at the World Equestrian Games.
• December: Nicholson dumped from High Performance Squad after officials complained about his behaviour.
• January: Equestrian NZ writes to Nicholson setting out behavioural conditions of a possible return to the elite squad.
