Kirkby, who was CEO of Telia rival Tele2 between 2015 and 2018, was the board's final candidate, the newspaper said.
A decision is expected shortly, it added. Telia declined to comment on the report.
Telia's previous CEO Johan Dennelind, who took the helm in 2013 and oversaw growth include a push into television - unexpectedly resigned in August.
Telia, in which the Swedish state owns 38%, proposed last week to replace chairwoman Marie Ehrling with Lars-Johan Jarnheimer, who served as CEO of Tele2 between 1999 and 2008 and is chairman of publishing group Egmont Media.
(Reporting by Helena Soderpalm; editing by Jason Neely)