* CEO Bejar to step down by Sept 30
* FCC has hired head hunters for new CEO
* Move could lead to new rights issue and strategy (Adds details and background)
MADRID, July 13 (Reuters) - Spanish builder FCC, whose top shareholder is Mexican billionaire Carlos Slim, said on Monday that it could accelerate the departure of CEO Juan Bejar if it finds a successor before Bejar is due to leave at the end of September.
It said it had hired head hunters to find a replacement for Bejar, whose resignation was reported by Reuters last week.
His departure will enable Slim to increase his influence over the indebted company and could open the door to another rights issue and strategy change, sources had previously said.
Bejar has accepted an 8.4 million euro ($9.2 million) severance payment. He had signalled he would resign after Slim increased his stake via a rights issue in December.
FCC has pledged to reduce its debt to EBITDA ratio to less than four times, from 6.2 times at the end of 2014, but it has continued to suffer due to the weakness of the Spanish construction sector. ($1 = 0.9074 euros) (Reporting by Tracy Rucinski; Editing by Julien Toyer and Susan Fenton)