No.
Restoring the old Concorde to flight status would mean getting a reliable part supply and re-establishing the airworthiness certificate, and that would cost about as much as certifying the aircraft from scratch.
As to designing a replacement successor aircraft capable of supersonic speed, the answer is again NO for now.
Just look how much the 787 project cost: $32 billions (see link); that for an aircraft that was all new (same as would be required for a new supersonic), but still of a rather conventional subsonic design. How much would a supersonic of comparable capacity and range to a 787 would cost to develop? Conservative estimate would be 3 times as much.
Now, the 787 has sold a bit over 1000 units (most still to be delivered) in 10 years.
How many supersonic aircraft with the same basic mission would be selling?
For the m get go, less than half. Why? A supersonic flies faster, so can carry more people in less time. At twice the speed of sound -- which is the minimum speed for a supersonic to be really competitive, and was thus the speed the Concorde was cruising at -- a plane would be able to fly both ways on a typical route, and would therefore replace two slower aircraft.
But the interesting thing is that, if the supersonic design, development and certification cost is amortized on half as many plane, each plane would therefore have to carry twice as much of a premium, above and beyond the part and assembly cost, to account for it. So, each supersonic would then be priced higher still because it would sell in lower number.
So, flying supersonic would be costlier right off the bat, not even taking into account the cost of fuel, which will have to be a lot higher (drag increases with the square of the speed, so flying twice as fast would require 4 times as much fuel per unit time; and since you are flying twice as fast, so half as long, a supersonic would take twice as much fuel as a subsonic airplane; not taking into account the other aspects that bring a degree of inefficiency, like the airframe having to be heavier per passenger, to resist the heat and stress of flying faster), so twice as much fuel is a very conservative estimate.
So, we already have a plane that would cost DOUBLE as a very minimum, to operate, and three times more to develop and therefore buy.
Now, let's compound this with the desirability factor. Suppose that you fly regularly, say, between Chicago and London. You have the choice of a subsonic aircraft flight (a quick google search shows the non-stop between Chicago and UK ranging between $2574 and $3555) or a supersonic flight that would be between $7500 and $10000. How often would you be flying supersonic? One time out of 5? Once out of 10?
Imagine that the same frequency applies to everyone, and suddenly the market for the supersonic is again shrinking 5 or 10 fold, since the slower but much cheaper subsonic aircraft would still be there.
So, now the market for a supersonic that would fly mission similar to a 787 is reduced to 10 airplane sold per year, compared with 100 for the 787.
Which means even less plane to amortize the development costs. Which means each plane will sell for more. Which means it will be less competitive, with still higher ticket cost. Which means it would sell even less.
So, at the bottom line, you have the likes of Boeing and Airbus who would love to develop a supersonic (trust me, I am an aerospace engineer, we would just LOVE to work on a SST), but are unwilling to come up with the $100 billion up front to develop it, unless the airlines are so sure they will buy it, they would be willing to put money up front to reserve a delivery slot of the aircraft. Which the airlines will not do unless they have money, which they don't.
Unless future passengers are willing to pay their tickets now for flight 15 years form now.
Are YOU willing to invest $10000 right now for the ticket on a flight 15 years from now? And if you are, how many more people are as willing to finance the project?
Let's have a show of hand...
Fly faster than the Concorde did, and you increase all those costs dramatically, and sell even less plane, both because more expensive planes are harder to sell, and because faster plane are more productive, carrying more people in the same time, shrinking their own market.
That is why there will not be a successor to the Concorde in the currently foreseeable future. Something massive would have to occur first to make it plausible.