Prop Vunipola suffered a setback in his hamstring injury and was absent as England defeated Italy 37-0 at St James' Park on Saturday (AEST) in a final warm-up match ahead of the tournament in Japan.
Jones's men led 9-0 at half-time thanks to a trio of Owen Farrell penalties, but Ben Youngs, Joe Marchant, Ellis Genge and Anthony Watson touched down in the second half as Italy was put to the sword in Newcastle.
England next faces Tonga in its opening Pool C game in Sapporo on 22 September and takes on United States four days later, meaning Saracens forward Vunipola is expected to return against either Argentina or France.
"He's probably going to be right for the third or fourth game," Jones said.
"He just had a little scar tissue which was impairing his movement. But that's all been fixed. He's progressing really well now."
On whether he was worried about taking an injured Vunipola to the Rugby World Cup, Jones added: "No, not at all. He's a very important player for us.
"We feel that he's going to be ready to contribute in the World Cup. We've got two looseheads with us who are in great form, one who can play tighthead, so we've got three tightheads as well.
"We're confident that he'll make a great contribution for us."
Exeter Chiefs wing Jack Nowell will also miss England's first two Rugby World Cup games after dealing with an ankle injury and appendicitis.
"Jack Nowell is probably in the same category [as Vunipola]," Jones added. "They are important players for us, so we are prepared to be patient with them."