The letter, a copy of which was obtained by NBC News, was dated Thursday and comes after allegations emerged that Trump demanded in a meeting with lawmakers this month to know why the USA should admit immigrants from "shithole countries", including African nations.
"Now is the best time to bring your money, your jobs, your businesses to America", he said, singling out tax cuts and curbs to regulation as boosting the investment climate.
The letter, dated Thursday, is addressed to African leaders as they gather for an African Union summit this weekend in Ethiopia's capital Addis Ababa.
Mr Trump has denied using that language, but others present insist he did.
Mr Trump's letter, seen by The Associated Press and confirmed by two United States officials, says the U.S. "profoundly respects" the partnerships and values shared by the USA and Africans and that the president's commitment to strong relationships with African nations is "firm".
On Friday U.S. President Donald Trump took his "America First" message to the world's elite at Davos, telling a summit of business and political leaders that the United States would "no longer turn a blind eye" to what he described as unfair trade practices.
Noting South Africa's legacy of the racist system of apartheid, Mohale said in the open letter to Trump, "Many of us will be boycotting your address to delegates at Davos in protest against your divisive comments and continued failure to unequivocally apologize".
The letter gives no details on Tillerson's upcoming visit.
For his part, Kagame tweeted: "Had very good bilateral meeting with @realDonaldTrump!"
Neither side referred to Trump's vulgar comment.
The 55-member continental body criticised the U.S. president's remarks earlier this month.
The US will be represented at the summit by ambassador Mary Beth Leonard, US envoy to the AU, instead of the traditional high-level delegation.
An AU spokeswoman has said the organization was "frankly alarmed" by Trump's remark, and a number of African nations have spoken out or summoned USA diplomats for an explanation.
The letter offers President Trump's "deepest compliments" to the African leaders as they gather. "When the United States grows so does the world".