The Thai player is on five under, one shot behind leader Ana Menendez of Mexico.
Canada's Brooke Henderson holds the current record, winning an event on the Canadian Women's Tour in June 2012 aged 14 years, nine months and three days.
Thitikul turned 14 on 20 February.
Henderson's record for a male or female player came in a 36-hole event, bettering the mark by two days set by New Zealand's Lydia Ko in January 2012 at the New South Wales Women's Open.
Ko holds the current record on the Ladies European Tour, having won the 2013 New Zealand Open aged 15 years, nine months and 17 days.
Thitikul opened with a two-under round of 70 at Phoenix Gold Golf and Country Club in Pattaya, hitting 71 in round two and matching playing partner Menendez with a 70 on Saturday.
Wales' Amy Boulden, 23, shot a five-under 67 on Saturday - the lowest round of the tournament so far - to move into a three-way tie for third on two under.
Fourteen-year-old amateur Atthaya Thitikul is in contention to become the youngest known winner of a professional golf tour event after day three of the Ladies European Thailand Championship.