The sources said Deutsche Post's supervisory board is due to meet on Wednesday and Deutsche Telekom's board will meet a week later to discuss the matter.
Both companies declined to comment.
The Handelsblatt newspaper reported on Saturday that Appel would potentially be proposed for election at Deutsche Telekom's annual meeting on April 7.
The term of office of Telekom chairman Ulrich Lehner, who has headed the Telekom supervisory body since 2008, ends at next year's shareholder meeting. He had already confirmed that an external search for a successor was under way.
Appel's predecessor at Deutsche Post, Klaus Zumwinkel, also served as supervisory board chairman of Telekom.
The German government holds stakes in both companies.
Appel, a former McKinsey consultant, has been with Deutsche Post since 2000. In 2002, he became a member of the board of management, and in 2008 he moved up to the post of CEO.
His contract runs until 2022 and a decision on his future at the Post had been expected soon. Some industry insiders have speculated that Appel could be ready to move on given that Deutsche Post has posted record results through the pandemic.
(Reporting by Matthias Inverardi and Nadine Schimroszik; Writing by Emma Thomasson; editing by David Evans)