Bankia is under pressure to raise funds after asking for a €19 billion (£15.3bn) lifeline from the Spanish state last month. Its stake in IAG is worth around €419 million at current prices.
In an interview with a Spanish newspaper, Walsh said: “There are investors interested in replacing Bankia and, in my opinion, it is not a question of if, but when they are going to sell.”
He said he was open to other airlines becoming shareholders in IAG as long as they remained “neutral”.
IAG was formed by the merger of British Airways and Spanish carrier Iberia in 2010. The group posted a pre-tax loss of €263m for the three months to 31 March, up from €47m a year earlier, after it was hit by higher fuel costs and the economic crisis in Spain.
Following its recent takeover of BMI, the group sold the Bmi Regional business to a consortium of Aberdeenshire businessmen for £8m last month, saving more than 300 jobs in the area.