The offer values the firm at $715m Australian dollars ($524m; £347m).
Last December, Transfield Services rejected an offer from Ferrovial that valued the firm at about A$1bn.
Monday's takeover bid is for $1.35 per Broadspectrum share.
The takeover offer was made by a local subsidiary of Ferrovial and saw Broadspectrum's shares rise more than 50% on the news.
Broadspectrum later asked its shareholders not to take any action and said it would advise them of its views "shortly".
Transfield Services changed its name to Broadspectrum after its privately-owned former parent firm, Transfield Holdings, withdrew its permission for the detention centre operator to use its brand.
The company operates in several sectors including infrastructure, property, defence and financial services.
It employs more than 25,000 people and has contracts to run Australia's offshore detention centres on Papua New Guinea's Manus Island and on Nauru in the Pacific. Both centres have come under criticism for the way they have been managed.
Ferrovia's chief executive Inigo Meiras said if their takeover offer was successful, it would represent "a solid step in Ferrovial's strategy to expand its global footprint and the group's presence in Australia".
"Ferrovial has a proven track record of disciplined acquisitions and successful integrations," he added.
The offer is subject to approval by Australia's Foreign Investment Review Board.
Spanish infrastructure firm Ferrovial SA has made an all-cash takeover bid for the Australia detention centre operator Broadspectrum - formerly Transfield Services.