Ishiba has about 34% of the public's support, more than double the 14% for Chief Cabinet Secretary Yoshihide Suga, the second-most popular choice, a weekend Kyodo News survey showed.
A Nikkei/TV Tokyo poll showed Ishiba with 28% support, followed by current Defence Minister Taro Kono with 15%.
Suga came in fourth place with 11%, the poll showed.
The surveys highlight a split between public opinion and internal Liberal Democratic Party politics.
Suga - a longtime lieutenant of Abe's in a key supporting role - is expected to get the backing of the faction led by LDP Secretary-General Toshihiro Nikai and of other major factions, local media reported, putting him in a favourable position.
That would make it an uphill battle for Ishiba, a vocal Abe critic who unsuccessfully challenged the out-going premier in the last LDP leadership race in 2018 and is considered less popular within the party.
Another potential candidate, LDP policy chief Fumio Kishida, came in last place in both of the public opinion surveys.
Abe's resignation announcement on Friday, citing the worsening of a chronic illness, set the stage for the party leadership election, which public broadcaster NHK said was likely to place around Sept. 13 to 15.
The LDP president is virtually assured of being prime minister because of the party's majority in the lower house of parliament.
Brad Glosserman, deputy director of the Center for Rule-Making Strategies at Tama University, said Suga was the safe bet in terms of internal LDP dynamics, but might not be ideal come election time. A general election must be held by late October 2021.
"He doesn't seem to have either the charisma or the vision to push Japan in a new direction. He seems to be the eternal Number Two - he delivers on promises made by his boss," said Glosserman.