Mr Brunson is standing trial in Turkey over terrorism charges, which lies at the heart of a diplomatic crisis between Ankara and Washington and has prompted a crash of Turkey's lira currency.
According to the Hurriyet newspaper, which published Mr Erdogan's comments, the president told reporters that Turkey followed the rule of law and the US would not be able to make progress in the case by using threats.
Brunson who was working as a missionary, was arrested in October 2016 and later charged with links to a failed plot to overthrow the Turkish government in July 2016. Prosecutors are calling for a 35-year prison sentence in his case.
Supporters claim he is effectively a political prisoner being held as a bargaining chip by Turkey as it seeks the extradition of preacher Fethullah Gülen, which it blames for the failed coup.
Tens of thousands of people have been arrested and lost their jobs following the failed coup attempt.
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