1. Definition of an organometallic compound

   An organometallic compound is one that contains at least one metal–carbon bond where the carbon belongs to an organic ligand such as CO, alkyl, aryl, etc.

2. Analyze the compound

   The given compound is: $Mn_2(CO)_{10}$

   Here, carbon monoxide ($CO$) acts as a ligand and binds to manganese through the carbon atom, forming an $Mn-C$ bond.

3. Check the options

   • A: $C-O$ bond
     This bond is present in carbon monoxide itself, but it does not make a compound organometallic.

   • B: $Mn-Mn$ bond
     A metal-metal bond does not define organometallic character.

   • C: $Mn-O$ bond
     In metal carbonyls, bonding occurs through carbon, not oxygen.

   • D: $Mn-C$ bond
     This is the defining feature of an organometallic compound.

4. Conclusion

   Therefore, $Mn_2(CO)_{10}$ is organometallic due to the presence of the $Mn-C$ bond.

Correct option: D