Former Republic of Ireland youth team player Brennan, who joined the club in 2013 at under-14s level and signed his first professional contract with the Owls in May 2018, will stay with Notts until November 27.
Brennan made his senior Wednesday debut in September 202 at Rochdale in the Carabao Cup and has made five appearances for the club, with his only outing this term coming in the recent Papa John's Trophy game with Newcastle United under-21s.
Brennan is the second Owls player to head to the National League with Alex Hunt linking up with Grimsby Town on a half-season loan last month.
Left-back Ryan Galvin, 20, joined Conference North side Gloucester City in the summer in another temporary arrangement, until the start of January, while goalkeeper Luke Jackson, 19, is currently on a month's loan at Guiseley.
Midfielder Liam Waldock, 20, is currently on loan at Northern Premier League outfit Gainsborough Trinity, with striker Charles Hagan, also 20, part-way through a half-season loan at Conference South club Hampton and Richmond, which lasts until January.