The Guardian says Celtic and Tottenham see Keane as a long-term managerial option. Both clubs currently have caretakers in place in Martin O'Neill and Igor Tudor respectively.
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Keane has worked in both Israel and Hungary, where he led Maccabi Tel Aviv and later Ferencváros.
Ferencváros won the Hungarian title last season and also top the table ahead of this weekend's matches.
Celtic are already working on a deal to bring Keane to Scotland at the end of the season, while Tottenham have their former striker on their shortlist of options for next season.