Levante won their Group L opener at home to Helsingborg 1-0 but then lost 2-1 away to Hannover, and Barkero is keen to make the Estadio Ciutat de Valencia something of a fortress in order to reach the knockout stages.
"If we want to go through, at home we have to be very strong," he said.
"If we win all three of our home games, we would be very much on track."
Back-to-back wins in the Primera Division have lifted Levante to seventh place, and Barkero wants to keep the momentum going.
"The team has a good dynamic and we must take advantage in the Europa League," he said.
"To get to six points would give us a good chance to get to the next phase, but we know the teams in our group are very evenly matched."
FC Twente arrive in Valencia as the Eredivisie leaders, although their advantage was trimmed to one point at the weekend as they were held to a 1-1 draw away to Roda JC, allowing PSV Eindhoven to close the gap.
Although they did not lose, you might not have thought it from Steve McClaren's post-match comments as he left his players with no doubt that they will need to do better to meet his standards on Thursday night.
"We did not start well in this match, our place was too slow," McClaren said.
"We lost the ball too much and did not control the game. We played too often through the middle and did not find space.
"We will have to learn for next time that we need to focus instantly at the start and not just after 45 minutes."