The Irish side have announced that they will challenge the decision at the Court of Arbitration for Sport (CAS).
Drogheda qualified for the European competition by winning the FAI Cup in 2024.
However, Silkeborg, a Danish team also owned by the American Trivela group, also secured their place in the Conference League, thanks to a play-off win after finishing in first place in the play-out phase.
Drogheda, meanwhile, finished ninth in the Irish Premier League.
UEFA regulations prohibit two clubs controlled by the same owner from competing in the same competition at the same time.
Under UEFA's criteria, the higher-ranked team - in this case, Silkeborg - is prioritised, which automatically leads to Drogheda United's exclusion.
'We are devastated'
In a statement, Drogheda on Thursday said: "The club can confirm that we have been notified by UEFA that we are facing exclusion from the UEFA Conference League, due to the fact that both Drogheda United and Silkeborg have qualified for the same competition.
"This is despite significant efforts by the club to work constructively with UEFA to realize the necessary ownership and governance changes to allow both teams to participate in the last few months.
"We are devastated by this potential decision and believe that it would be both painful and unfair, given the way in which the new rules introduced this year have, in our view, been communicated and applied unevenly between different clubs."
"We are challenging the decision at the Court of Arbitration for Sport.
"We have been in an active dialogue with UEFA for months and have proposed share disposals, fiduciary arrangements and other commitments in line with recent CFCB precedents - all of which have been rejected. However, we look forward to being heard by CAS and thank you in advance for your support."
The exclusion is a major financial blow for Drogheda, which could have earned over €500,000 from participation in the Conference League alone.
UEFA has also confirmed that no other Irish Premier League side can replace Drogheda, as the deadline for the club to obtain their European licence has expired.