Messi has agreed in principle to acquire all the shares held by majority owner TH Soluciones Group S.A.S., with an announcement expected in the coming days, the reports said.
The Colombian investment group, which took control of Eldense in October, 2025, would transfer its entire stake to Messi. The deal remains subject to the approval of Spain's High Council for Sports (CSD).
Reuters has contacted TH Soluciones and Eldense for comment.
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Messi, who recently retired from international duty with Argentina, acquired fifth-tier Spanish side UE Cornella on April 16.
He is also due to receive a stake in Major League Soccer club Inter Miami, once his playing contract with the team ends.
Eldense are based in Elda, a town of about 55,000 people in the eastern Spanish province of Alicante.
The club have spent most of their history in Spain's lower divisions but, following fresh investment, have played in the second tier in three of the past four seasons.
Eldense sacked manager Claudio Barragan on Sunday after taking two points from their opening four league matches, leaving them third from bottom.