Another Taliban leader added: "When we heard about Trump being Covid-19 positive, we got worried for his health, but seems he is getting better."
Tim Murtaugh, Trump's campaign spokesman, said on Saturday that they "reject" the endorsement.
"The Taliban should know that the president will always protect American interests by any means necessary," Murtaugh added.
There are fewer than 5000 US troops in Afghanistan. That number is expected to drop to 2500 by early next year, national security adviser Robert O'Brien has claimed.
In February, the White House and the Taliban signed a pact where they planned to have troops withdrawn by the spring of 2021. The Taliban is required to split from al Qaeda and negotiate a deal for sharing power with Afghan government rivals.
Trump announced last week that all troops should be "home by Christmas", almost in exact contrast to the timeline that most US military commanders have.
"We should have the small remaining number of our BRAVE Men and Women serving in Afghanistan home by Christmas," he tweeted.
The Taliban has expressed support for Trump's "America first" stance.