Goal understands the Championship leaders saw offers for Gomez rejected during the summer, but with his contract set to run out at the end of the season, Leeds are confident a deal can be agreed in early 2019.
The 28-year-old scored two goals during the European campaign that year, though Athletic eventually came up short, losing to Atletico Madrid in the final.
Since then set-piece specialist Gomez went on to become a regular in the Athletic line-up before moving to Alaves on a free transfer in the summer of 2016.
He played his part in their escape from relegation during the second half of the 2017-18 campaign and has started the current season in fine form, scoring three goals in nine games.
Nevertheless he has continued to impress, with Alaves having recently beaten Real Madrid - a result which has helped them climb to third in the table, just a point behind leaders Barca.
In total Gomez has scored 17 goals and laid on 10 assists in 82 appearances for Alaves, but Bielsa is hopeful he can tempt him away from Spain to West Yorkshire.
Bielsa has been a revelation at Elland Road so far this season, with his side having climbed back to the top of the table on Wednesday evening with a 2-0 win over bottom-of-the-table Ipswich Town.
The Argentine broke the Leeds record for the most successive wins at the start of his tenure, and though their form has fluctuated somewhat since, they remain well in the hunt for a return to the Premier League for the first time since 2003.
Bielsa and Leeds are likely to face competition from a host of Spanish clubs for Gomez while Alaves have not given up on him agreeing a new contract to remain at Mendizorrotza.