Oceania Football Confederation (OFC) president Temarii, of Tahiti, is also alleged to have asked for a payment, in his case to finance a sports academy.
Blazer said all bidding countries he had dealt with had behaved entirely properly.
He added: "I'm disappointed with what I have read but you can't say the system is bad. They have created a scam, a trap, tempting people to do something wrong and it's up to the Fifa ethics committee to make their recommendation."
Blatter has written to the 24 executive members promising a full investigation.
Fifa has the option of postponing the 2 December vote, which will decide the host of both World Cups.
But Blazer said: "We should deal with it within the timeframe established. We want to keep the issues separate and it's important we conclude the World Cup decision.
"There is no reason why we shouldn't. The investigation can be done right away."
England, Russia, Spain/Portugal and Netherlands/Belgium are bidding for the 2018 World Cup while the United States, Australia, Qatar, Japan and South Korea are campaigning to host the 2022 tournament.
"I have met with five different bidding committees and in some cases visited their countries - England, Russia, Belgium/Holland, Japan and the United States," added Blazer.
"They were all very professional and proper and not alluding to any other types of benefits and that needs to be said. They were simply trying to offer the best World Cup proposals."
The OFC says it is investigating the reports.
"Further to information made public by the Sunday Times, the OFC president and Fifa vice president Reynald Temarii will cooperate fully with the Fifa Ethics Committee and the Fifa secretary general," a statement read.
"Reynald Temarii welcomes a full and thorough investigation so that all the facts can be heard."
England and the US initially asked to be considered as hosts for either the 2018 or 2022 World Cup.
But on Friday, the Americans withdrew from the 2018 race, while England did the same for 2022.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.