Ratnakar Shetty, chief operating officer for the Board of Control for Cricket in India (BCCI), told the BBC: "In principle [the England team] do feel that the tour has to go on but I think a clear picture will emerge by [Monday] evening or tomorrow morning as to how we will proceed in this matter.
"I think a team which visits and then took a break and went back - when it comes back I think it sends a great deal of positive signal to the world."
Last week's terror attacks in Mumbai saw at least 172 people killed.
With the last two one-dayers between India and England already cancelled, the two-Test series depends on the security report's outcome.
It is understood that travel will be limited as much as possible, with England flying directly to Chennai to start the tour.
ECB managing director Hugh Morris could not confirm reports of Andrew Flintoff and Steve Harmison refusing to travel.
"I've not had confirmation from either Andy or Steve," Morris told BBC 5 Live's Sportsweek programme on Sunday.
"What we are doing is pulling information together over the next 24 to 48 hours and we will be putting that to the players and their representatives."
Morris said the team would only go back to India if their safety could be guaranteed, but it remained unclear as to which players a returning England team would.
He said England could send a weakened squad if regular first-teamers declined to go, conceding: "Clearly we would have to potentially face that issue."
I'm sure families will want to influence players and rightfully so, but they need to make their own decision
Lalit Modi, Indian cricket board
Swann told BBC Radio 5 Live's Sportweek programme: "I wouldn't be 100% happy going back. If [the report] says 'look guys we're safe, we can go out there, we can do some good playing cricket,' then I'd go.
"But until that report comes back, I'm not keen at the minute."
Meanwhile, Lalit Modi, vice-president of the Indian cricket board (BCCI), expects England to return - if safety assurances are met.
"Player safety and fans safety is paramount for us," Modi said.
"But on the other hand we must make sure events like this don't give the opportunity to terrorists around the world to get away with this. We need to carry on.
"I'm sure families will want to influence players and rightfully so, but they need to make their own decision.
"I really do believe it will happen though. The ECB, the security teams and governments are all checking it out and I'm sure it will be all right."
He also said the BCCI would not press the ECB for compensation should the series be shelved.
International Cricket Council chief executive Haroon Lorgat urged the series to go ahead if it was safe to do so.
"If safety and security allow then I would urge the England Test tour of India to go ahead," he said, "and if it does so, then representatives of the ICC will be there to show solidarity with the competing teams."
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