Blatter, 75, was re-elected as president of the world governing body for a further four-year term on Wednesday against a background of corruption scandals.
Johansson, 81, who headed the European governing body for 17 years, made his comments in an interview in a news programme on the Swedish national broadcaster SVT.
"He came to me, someone who was close to me, that I co-operated with for several years previously, who had seen how brown envelopes were given from one to the other," he said.
"The interpretation was that it was to get them to vote for Blatter. No proof was forthcoming so I decided not to refer to it, because that could put me in an equally bad position, just by believing rumours and making accusations that weren't proved."
Johansson also said that Blatter's plan to allow the FIFA congress to decide which notions would host future World Cups was effectively a vote of no confidence in the FIFA Executive Committee.
"I think he's disapproving his own executive committee, his nearest (colleagues)," he said.
"It's they who, before they make any decision, have to make sure that they have all the details, on finances, on security, on communications, on hotels, on everything and have a clear picture of who the best candidates are."
