Competition concerns and broadcasting standards were among the issues of concern, she said.
Ms Bradley has asked for evidence from the two companies and will decide later this month whether to intervene.
Both Sky and Fox are controlled by businessman Rupert Murdoch, who also owns the Times and the Sun newspapers.
A decision to intervene would not block the deal, but trigger an Ofcom assessment as well as a Competition and Markets Authority report to be considered by Ms Bradley.
21st Century Fox said it was confident the deal would be approved: "We anticipate regulators will undertake a thorough review of the transaction and we look forward to engaging with them."
The Culture Secretary said she had written to the two companies to inform them of "concerns that there may be public interest considerations ... that warrant further investigation".
The first public interest ground was media plurality and the need for a "sufficient plurality of persons with control of the media enterprises serving audiences in the UK".
Commitment to broadcasting standards was the second ground for possible intervention.
This related to the need for those controlling media enterprises to "have a genuine commitment to attaining broadcasting standards objectives".
The proposed deal has also been lodged with the European Commission, which would examine competition concerns if the case was referred by Ms Bradley.
Rupert Murdoch's 21st Century Fox is offering £11.7bn for the 61% stake in Sky it does not already own.
Sky shareholders would receive £10.75 in cash for each share, valuing the entire company at £18.5bn.
Shares in Sky fell 2p to 996.5pp in London, giving the firm a market capitalisation of £17.1bn.
Culture Secretary Karen Bradley said she is "minded" to order an Ofcom investigation into the planned £18.5bn takeover of Sky by 21st Century Fox.