Horford has entered health and safety protocols, while Smart is still dealing with a right mid-foot sprain.
Celtics coach Ime Udoka said he initially found out about his status around 5 p.m. ET on Tuesday, or roughly 3 1/2 hours before tip-off. He would not say if Horford tested positive.
"As always, we don't comment on the status of our guys,'' Udoka said. "He's feeling fine. We'll go from there. Wait to see results and tests and future tests.''
Typically, it has taken players who test positive a minimum of five days to escape the protocols. That would bring Horford's availability for Game 2 in Miami on Thursday into question -- and possibly longer. The teams will play every other day in this series; the Celtics will play host to Games 3 and 4 on Saturday and Monday.
According to ESPN's Adrian Wojnarowski, the Celtics are prepared for the "likely scenario" that Horford remains in protocols for Game 2.
Wojnarowski also reports that big man Robert Williams, who has not taken the court since May 7 due to a sore left knee, should be able to play "significant minutes" on Tuesday.
Smart, the defensive player of the year, was listed as questionable earlier on Tuesday after suffering the injury during Boston's Game 7 win over the Milwaukee Bucks on Sunday.
It's not yet clear how long he is expected to be out. Results of the MRI on his foot came back clean, Wojnarowski reported Monday.
"The soreness was too much,'' Udoka said, adding that Smart is also dealing with swelling. Smart will continue receiving treatment and be re-evaluated Wednesday, Udoka said.
With both Smart and Horford out, the Celtics are down two starters to begin the Conference Finals.
Horford is averaging 13 points and 9.4 rebounds, shooting .524/.451/.833 this post-season. The 35-year-old broke out with 30 points to lead the Celtics in a crucial Game 4 win over the Bucks in Round 2.
Smart, who missed a game earlier these playoffs due to a quad injury, is averaging 15 points and 6.2 assists in 10 games.
Miami had already ruled point guard Kyle Lowry out for Game 1 because of a hamstring injury.
With files from the Associated Press