Projections by Sigma Conseil do not show Prime Minister Youssef Chahed among the top four candidates in Sunday's vote, which featured 26 contenders.
Official preliminary results are expected by Tuesday.
The projection gives Saied, a constitutional law professor without a party, 19.5% of the vote and Karoui 15.5%.
Ranking third is the candidate for the moderate Islamist party Ennahdha, Abdelfattah Mourou, with 11%.
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