Guido Tognoni worked "extremely closely" with the Fifa president during Blatter's second term.
"In Fifa, for many years, you could only reach your goal by taking dollars in your hands," Tognoni told BBC Sport.
Fifa said investigations into such claims were ongoing and, as yet, there was no evidence of wrongdoing.
The bidding process for the 2018 and 2022 tournaments is the subject of a Swiss criminal investigation.
Asked if countries had any choice other than to bribe Fifa during the bidding process, Tognoni replied: "This speculation is permitted, yes."
Russia and Qatar deny wrongdoing, while Blatter has not been named in a separate US criminal investigation into widespread corruption at Fifa.
Domenico Scala, head of Fifa's audit and compliance committee, said if evidence emerges from either the US or Swiss investigations which shows the result was affected by bought votes, the "awards could be cancelled".
However, Tognoni, who was Blatter's right-hand man for a period in the mid-90s and again between 2001-2003, said he did not think the World Cups would be taken away from Russia or Qatar.
"Even if there is evidence that Fifa people were bribed, where is the problem? With Fifa, or the people who had no choice but to get the World Cup with bribing?"
In response, Fifa said in a statement on Monday: "Fifa initiated the investigation by the Swiss authorities precisely to answer questions such as this.
"However, while investigations are ongoing, it should be noted that to date, no evidence has come to light to suggest there are any legal grounds for rescinding the current Fifa World Cup selections."
Seven senior Fifa officials were arrested last month in high-profile dawn raids at a Zurich hotel, where they had been staying before the presidential election.
They are among 14 people charged by US prosecutors, who allege that bribes and kickbacks of more than $150m (£100m) were accepted over a 24-year period.
Blatter, Fifa president since 1998, was re-elected at the Zurich meeting but announced four days later that he would resign amid the corruption allegations engulfing the organisation.
American lawyer Michael Garcia was previously hired by Fifa to investigate the 2018 and 2022 World Cup bidding processes, but he complained that the published summary of his 430-page report was an "erroneous" representation of his work.
The report cleared World Cup hosts Russia and Qatar of wrongdoing.
Garcia quit weeks later over Fifa's handling of his inquiry and his comments reopened the debate about the validity of the bidding process.
Scala has since had Garcia's report reviewed by legal experts, from both in and outside of sport, but neither were able to find grounds to alter the award of either tournament at this stage.
Russia and Qatar may have had to pay bribes to secure their World Cups, Sepp Blatter's former special advisor has suggested.