Donegal produced an unconvincing display in their round 2A win over Longford on Saturday to progress to the next round of the series.
The ties will be played on 8 July, with three games involving Ulster counties in round 2B having already scheduled for this Saturday.
Wexford play Monaghan, Westmeath face Armagh and Cavan host Tipperary.
Mayo ensured their passage by beating Derry in extra-time as the Oak Leafers became the third Ulster county to exit this year's All-Ireland series, following earlier defeats for Antrim and Fermanagh.
Meath were beaten by Kildare in the Leinster SFC and edged past Sligo on Saturday, but they may prove a difficult obstacle in Navan for a seemingly out-of-sorts Donegal.
Monday's draw for the second round of the All-Ireland hurling qualifiers paired Dublin with Tipperary and Waterford with Kilkenny.
Donegal have been drawn away to Meath in Round 3A of the All-Ireland Football Qualifiers, while Clare will host Mayo.