Strache was shown in the footage of as-yet unknown origin meeting a woman posing as the niece of a Russian oligarch in 2017, shortly before the election that brought him to power.
In the footage published by German media on Friday evening, he appears to offer to funnel contracts toward a company in exchange for political and financial support.
“It was dumb, it was irresponsible and it was a mistake,” Strache told a news conference, fighting back tears as he asked his wife and others to forgive him. He maintained, however, that he had done nothing illegal.
Austrian Chancellor Sebastian Kurz is expected to make a statement later on Saturday as speculation swirled about whether the damage would be limited to Strache or if Kurz would call a snap election only a year and a half after the coalition was formed.
Kurz has repeatedly distanced himself from the FPO over lesser scandals to protect his popularity.
Strache said he would be replaced by Transport Minister Norbert Hofer, who narrowly lost 2016’s presidential election and is more popular than Strache.
“I do not in any circumstances want my wrong behavior to provide a pretext for the government to collapse,” Strache said.