I agree with MikeMI.
It is more complicated than you think.
If it is a regulated supply, then there won't be a problem.
But if it is unregulated, then the voltage with no load will be greater than 12 V, possibly 18 V or more.
If an unregulated supply is rated at 12VDC 1000mA, then it means that the voltage will be about 12 V when the current drawn is 1000 mA.
The voltage will be higher for lower currents.
So I suggest that you measure the voltge with no load.
If it is unregulated, then you need to know whether your linksys router can withstand the higher voltage. I think it can, but I can't be sure without seeing the circuit diagram.