The MSP was sent a threatening email which included a propaganda video from far-right extremists, the Daily Record reported.
It comes after Mr Sarwar spoke out about his experience of racism and Islamophobia while he was a candidate for the Scottish Labour leadership.
Mr Sarwar said he would not be deterred from campaigning against racism and Islamaphobia.
He told the Daily Record: “I have been overwhelmed by the messages of support and solidarity in the last few weeks. Sadly, I have also received some highly troubling communications.
“But all these people have inadvertently done is strengthen my resolve to campaign on the issue of everyday racism and Islamophobia and to deliver positive change.
“There are many who seek to divide our communities, to use perceived difference to build fear and resentment. Those that seek to divide must never be allowed to succeed.”
Police Scotland said they are investigating the email.
A police spokesman said: “We can confirm we are investigating a report of an offensive communication and inquiries are ongoing.”
Mr Sarwar has launched a cross-party group on tackling racism and Islamophobia at the Scottish Parliament.
The Labour MSP lost out to Richard Leonard in the leadership campaign, which was sparked by the resignation of Kezia Dugdale last year.
He claimed that during the leadership campaign a councillor told him he could not back him because he was a “brown, Muslim Paki”.
Mr Sarwar said he had been “inundated” with examples of everyday racism since revealing the allegation against Davie McLachlan, former leader of the Labour group on South Lanarkshire Council, who denies the claims and has been suspended by Scottish Labour pending an investigation.