You can't drain an aquifer without consequences
There are locations throughout the world today where people are suffering from declining access to water because aquifers are drained faster than they can recharge. The idea of greening the Sahara by, effectively, draining an aquifer might result in a green Sahara... but it will create deserts elsewhere.
The truth is that if we account only for current technology, each and every attempt to radically redistribute a water source has resulted in bad things. Take, for example, the Aral Sea. Formerly the fourth largest lake in the world and once the breadbasket for a region with a strong economy that included everything from food to recreation, the Soviets diverted almost all of the water from its sources to irrigate (aka, "green") Uzbekistan. The irrigation project succeeded! But it drained the lake to about 10% of its former glory and caused all kinds of problems with toxic salinity.
A Worldbuilding Solution
There is plenty of water to do everything humanity can imagine... if you can just get the salt out of it and transport it efficiently. The end result might be a much cloudier planet... which means less sunlight on the ground... which might actually kill everything (that's be just our luck...). But let's ignore all that.
Invention of the day!
I propose a practical fission/fusion/antimatter/whatever power plant big enough to provide energy for disintegration-level recycling and in need of a ferocious amount of water to keep it both cool and under control.
Sea water... and a honking lot of it... is brought in to cool the system. The process wonderfully results in desalinated hot water, which is pumped to massive cooling lakes (not ponds... lakes...) creating an entirely new ecosystem with tons of available water.
Place these power plants strategically so that the water is captured efficiently as lakes, creating freshwater rivers and streams back to the ocean.
And thus we have a green Sahara. Oh, it would take forever to get the Sahara to green up. The world is full of lakes and reservoirs (e.g., Utah's Strawberry Reservoir) that are full of water and surrounded by nothing more green than sagebrush... (After fifty years there's still nothing more green than sagebrush around Strawberry Reservoir.) The point I'm making is that water is only one of a bunch of things you need to get grass and trees — and deserts lack pretty much all of them.
But that's the subject of another question.
Ocean water + cheap desalination = green Sahara... eventually.