The MDC won majority in parliamentary elections and Tsvangirai claimed victory even in the presidential electioin.
But the country's election commission said he had not won enough votes for an outright victory, prompting the run-off vote.
Tsvangirai was expected back in Harare on Saturday to prepare to take part in the run-off vote scheduled for June 27.
Sibotshiwe said: "In light of this information [of a plot] ... it has been decided that the president will not return today.
"This will enable his security detail more time to analyse and mitigate the threat."
Zimbabwe's election process has seen repeated delays, violence and allegations of electoral fraud.
Tsvangirai accuses Robert Mugabe, Zimbabwe's president, of delaying the run-off to mount a campaign of intimidation against his supporters.