Don't believe that they are in no rush, they probably want to have a replacement for Pioli soon, but just haven't found one yet. And "No", I don't think it's likely they'll want to see the managers at the EURO, there aren't any great managers there who could possibly take their team to the top and then leave like that.
So far, based on rumors, there are 6 managers who might go to AC Milan, including: Paulo Fonseca, Roberto De Zerbi, Lionel Scaloni, Thomas Tuchel, Sergio Conceicao and Xavi Hernandez. Which of these managers do you think is most likely to go to AC Milan?
AC Milan have officially parted ways with Stefano Pioli this season after failing to present at least one trophy to AC Milan, but I think Stefano Pioli dismissal may not be just a reason for failing to win trophies this season, there may be a bad relationship between the management and Stefano Pioli so that they do not wait for the end of the season or when they have reached an agreement with another coach to sack Stefano Pioli this season, so far, actually Milan under Stefano Pioli coaching is not too bad because in fact Stefano Pioli managed to bring Milan to the scudetto in 2022 after previously the last time they got the 2011 season, besides that in the 2022/23 season he managed to bring Milan to the semifinals of the Champions League after previously they continued to fail to compete in the Champions League, this season is Stefano Pioli last season in Milan but he managed to bring Milan to finish as runner up Serie A.
To be honest, I think the dismissal of Stefano Pioli by Milan does not have a strong reason and will actually be a disadvantage for Milan to fire a coach who is reliable and also genius in concocting game strategies, currently there are several coaches who are indeed linked with Milan next season, but among these names it seems that Fonseca will indeed be a strong candidate to replace Stefano Pioli position.