The potential deal, which one of the sources familiar with the matter said could be worth as much as $500 million, comes as relations between regional rivals the United Arab Emirates and Turkey show signs of improvement.
One of the sources said the talks were at an advanced stage and a second said the pair were engaged in due diligence.
"As a matter of course, we do not comment on rumour or speculation," an Aramex spokesperson said.
MNG Kargo declined to comment.
Aramex, in which ADQ bought a 22.25% stake last year, announced this week it has split its core businesses to capture growth in the post-COVID-19 transportation and logistics industry.
As part of the reshuffle, it has created a new regional structure and appointed a chief strategy officer to head its international expansion and merger and acquisition plans.
Established in 2003, MNG Kargo is one of the leading courier services companies in Turkey, owned by private equity firm Turkven and the Sancak family since 2017. The company has over 850 branches and more than 11,000 employees.
Turkey's President Tayyip Erdogan held talks with UAE National Security Adviser Sheikh Tahnoun bin Zayed al-Nahyan last month which included economic cooperation and UAE investment in Turkey.
Abu Dhabi conglomerate International Holding Co, of which Tahnoun is chairman, is seeking investment opportunities in Turkey, its chief executive officer told Reuters last month.
While political differences between the two countries run deep, Ankara has moved to ease tensions with several Arab powers over the conflict in Libya, internal Gulf disputes and rival claims to Eastern Mediterranean waters.
An Emirati official told Reuters this month the Gulf state was interested in reinforcing ties, pointing to trade and investment opportunities in transportation, health and energy.
(Reporting by Davide Barbuscia in Dubai and Ebru Tuncay in Istanbul;Editing by Elaine Hardcastle)
By Davide Barbuscia and Ebru Tuncay