The overture comes as Trump's tariffs and repeated talk of making Canada the 51st state have pushed Ottawa to seek closer economic and security ties with Europe.
Carney has called for a "unique alliance" with the EU as Canada seeks to reduce its heavy dependence on US trade.
"I would like to work with you on opening the door for Canada being the first associate member of the EU," von der Leyen told Carney during her annual State of the European Union address in Strasbourg on Wednesday.
Carney received a standing ovation from European MPs.
He is due to address the European Parliament on Thursday.
Von der Leyen said Canada and the EU would build on their existing free trade agreement, known as CETA, with a broader "Alliance for the Future" aimed at creating "a common prosperity and economic security space".
She cast that closer relationship as rooted not just in trade but in shared democratic values.
"Above all, dear Mark, Europe and Canada believe in democracy," von der Leyen said.
"We believe that power does not belong to the strongest, the richest or the loudest, but to all of us. That democracies have the freedom to choose with whom to work."
Von der Leyen said the expanded partnership would cover advanced manufacturing, technology, defence production, the Arctic, energy, critical minerals, batteries, artificial intelligence, quantum computing, cybersecurity and economic security.
"This is a partnership not against anyone else, but for our common strength," she said.
"In short, we want to bring the relationship with Canada to the highest level possible."
