Neymar, 33, returned to Brazil after his ill-fated move to Al-Hilal in order to try to rediscover some form and fitness after years plagued by injury.
After initially impressing upon his return to Brazil, he is yet to provide a single goal contribution in his last five games across all competitions.
Amid speculation surrounding his future future, Santos president Teixeira told Radio Bandeirantes: "We are talking to Neymar’s team. It is a step forward, we are making a lot of progress in the new negotiations and we are very close to a new agreement to allow Neymar to stay for a new period.
"We hope it happens as quickly as possible. It can't be rushed. It's not a case of, if Neymar doesn't renew, he won't play tomorrow or Sunday. That's not the case. We have no need for Neymar to play now or for the team to return to action. That's my impression. I'm very hopeful for a favourable outcome."