I think there are two possible choices, either you must play 5 rounds of Euchre or 10 rounds - any other number between 1 and 14 rounds will lead to an unbalanced schedule where not all players get the same number of games, something that will no doubt lead to complaints.
For 5 rounds I think the following schedule is reasonable, the 5 pairs of players who have bye together never sit at the same table together.
( 7 6 v 2 8) (3 4 v 10 5) ( 1 9)
( 8 7 v 3 9) (4 5 v 6 1) ( 2 10)
( 9 8 v 4 10) (5 1 v 7 2) ( 3 6)
(10 9 v 5 6) (1 2 v 8 3) ( 4 7)
( 6 10 v 1 7) (2 3 v 9 4) ( 5 8)
For 10 rounds it is possible to arrange that all 8 partners that a player has are different and that everyone sits together at a table (either as partner or opponent) either 2 or three times.
( 1 4 v 7 3) ( 5 6 v 8 10) ( 2 9)
( 2 5 v 8 4) ( 6 7 v 9 1) ( 3 10)
( 3 6 v 9 5) ( 7 8 v 10 2) ( 4 1)
( 4 7 v 10 6) ( 8 9 v 1 3) ( 5 2)
( 5 8 v 1 7) ( 9 10 v 2 4) ( 6 3)
( 6 9 v 2 8) (10 1 v 3 5) ( 7 4)
( 7 10 v 3 9) ( 1 2 v 4 6) ( 8 5)
( 8 1 v 4 10) ( 2 3 v 5 7) ( 9 6)
( 9 2 v 5 1) ( 3 4 v 6 8) (10 7)
(10 3 v 6 2) ( 4 5 v 7 9) ( 1 8)
