Schulz is known to want to stay on as Parliament President until the end of the current legislature.
However, in order to do so he needs the support of the EPP group, the biggest grouping in the assembly, which is thought to favour a candidate from their group.
This has triggered speculation that Schulz may be tempted to leave Brussels and return to Germany in the new year.
One well-placed parliamentary source told this website, "It's not clear on Schulz yet. The general view is that if he can't get guaranteed support from the grand coalition then he won't run and will go off to German domestic politics."
Antonio Tajani, a former European Commissioner is current Parliament Vice-President, is believed to be considering throwing his hat in the ring to replace Schulz when members vote on the post in January. Tajani is a member of the EPP group.
Mairead McGuiness, an Irish EPP group MEP, is also being widely touted as a contender.
ALDE group leader Guy Verhofstadt is another name which had been cited as a possible candidate.
The Parliament source added, "The EPP group still can't decide on its candidate and Verhofstadt can't decide whether he's going to run."
The only declared candidate so far is Helga Stevens, a Belgian MEP and a Vice-Chair of the ECR group.
Her colleague Vicky Ford told this website, "I'm supporting Helga Stevens. A truly inspirational woman who breaks through glass barriers through her intelligence, care and hard work. A passionate fighter for equality."
Ford added, "I hope she will get many votes."
A spokesperson for the S&D group said it remained "firmly behind" Schulz, and while the deputy had the "full support" of his fellow Socialists, it was "up to the EPP group to decide what it is going to do."
An Italian Socialist group member who declined to be named said, "We firmly support a Schulz re-election. Of course, it is up to him to decide whether go for another term or not. The situation is very fluid and every option is still open."
Greens/EFA group deputy Bart Staes told this website, "I presume Schulz will go for another term. If so, the Greens will not support him. I heard there are already some EPP group candidates - Alain Lamasourre, Alojz Peterlee and Mairead McGuinness, as well as the ECR group's Helga Stevens.
"There will probably also be candidates from other groups. The Greens have not decided yet how they will tackle this election."
His Swedish colleague Linnéa Engström said, "I'd really like to see someone who puts all the parliamentary groups in focus, who does not just speak for themselves or defends the grand coalition.
"Obviously I would like to see a Green and a woman.
"I've heard rumours that Helga Stevens, Lamassoure, or McGuiness might stand. Stevens would not be a good choice, Morin-Chartier I could live with but there might be others interested. McGuiness could be a good choice since she is Irish and in the light of Brexit could function well in trying to bridge EU-UK dialogue."