In the other quarter final fixtures, Cardiff and Cross Keys will meet at Cardiff Arms Park after both claimed comfortable victories in the last round. The Blue and Blacks crossed for three tries in their 30-3 defeat of Newbridge while Keys got revenge for their recent league defeat to Bedwas by getting the better of the same opposition.
Another all Premiership quarter final sees Aberavon host Carmarthen Quins at the Talbot Athletic Ground with the final tie a meeting between Ebbw Vale or RGC 1404 and Llandovery.
On his side's last eight draw, Llanelli head coach Kev George said: "We are happy with what we've got. You have to beat the best sides if you want to win the competition and Pontypridd, if they go through, are certainly the best.
"They will be hot, hot favourites but hopefully we will be able to put a good side out on the day. We were a bit lucky in the last round against Bridgend Athletic but played some nice stuff and came through in the end. Hopefully that's a good omen for the rest of the season."
On facing either Ebbw Vale or RGC for a place in the semi-finals, Llandovery forwards coach Euros Evans said: "It's a really tough draw even though they are both Championship sides. You always want a home tie ideally but at this stage of the competition, there is no easy game.
"The league form suggests Ebbw Vale are the stronger side but we'll look forward to the game whoever we face. We played well against Tondu in the last round - we showed them the respect they deserved and picked a strong side.
"Everybody is looking ahead to the Millennium Stadium final but we're looking at it one game at a time at the moment."
In the SWALEC Plate, 13 teams remain in with a chance of reaching the Millennium Stadium. Last weekend's wet weather saw a number of games postponed with those fixtures needing to be replayed before the quarter finals are held next month. When they do come around, Amman United will travel to either Merthyr or Risca while Rhiwbina face a home tie with one of Caernarfon or Nant Conwy.
It's a similar story in the SWALEC Bowl, with only Hartridge HSOB, Ystradgynlais and Clwb Rygbi Cymry Caerdydd already guaranteed a quarter final berth. The other places in the last eight are still to be decided.
SWALEC Cup quarter final draw:
Cardiff v Cross Keys
Ebbw Vale or RGC 1404 v Llandovery
Aberavon v Carmarthen Quins
Pontypridd or Newport v Llanelli
SWALEC Plate quarter final draw:
Llangennech v Glamorgan Wanderers or Llantwit Fardre
Merthyr or Risca v Amman Utd
Rhiwbina v Caernarfon or Nant Conwy
Glynneath or Waunarlwydd v Kenfig Hill or Maesteg Quins
SWALEC Bowl quarter final draw:
Hartridge HSOB v Old Illtydians or Abertysswg Falcons
Llantwit Major or Beaufort v Pyle or Cwmgors
Tredegar Ironsides or Llandudno v Ystradgynlais
Clwb Rygbi Cymry Caerdydd v Cowbridge or Nantymoel
Ties to be played the weekend of March 22, 2014.