"The president is enjoying very robust health and is continuing to carry on his duties" a statement said.
It warned that people speculating on his health will be "brought to book".
Mr Mutharika came into office in May 2014.
He spoke at the UN on 25 September and has been out of Malawi since then.
The government statement, posted on Facebook, and signed by the government spokesperson, said there was no need for people to panic.
It added that the rumours "were baseless, malicious and sickening".
It also reminded "all Malawians that whilst the laws of this country provide for freedom of expression, those rights ought to be exercised responsibly and within the law.
"Spreading false rumours about the health of the president is a criminal offence."
Local activist Gerald Kampanikiza told the privately-owned Nyasa Times newspaper that the government should provide video evidence to disprove the rumours.
"Take a video feed of Mr Mutharika and post [that] rather than his last public appearance photo as you are doing," Mr Kampanikiza is quoted by the newspaper as saying.
Malawians online have been using the hashtag #BringBackMutharika to call for the return of their president.
Malawi's government has dismissed rumours about President Peter Mutharika's health following his extended stay in the United States after attending the UN General Assembly in New York.