Adding rivers to procedural generated 2D map

I’m using a noise function to procedurally generate a tile map, I was wondering if there is any way to extend what I’m already using to add the ability to create rivers to the map generation.

Currently I’m getting this:
map

And I’d like to find a method of doing this:
map with river
Where large bodies of water are connected to one another as if they were a single continuous flow.

I have a Noise class that makes the values for the map.

public class Noise {

	int seed;
	float frequency;
	float amplitude;

	float lacunarity;
	float persistance;

	int octaves;

	public Noise (int seed, float frequency, float amplitude, float lacunarity, float persistance, int octaves)
	{
		this.seed = seed;
		this.frequency = frequency;
		this.amplitude = amplitude;
		this.lacunarity = lacunarity;
		this.persistance = persistance;
		this.octaves = octaves;
	}

	public float[,] GetNoiseValues(int width, int height)
	{
		float[,] noiseValues = new float[width, height];

		float max = 0f;
		float min = float.MaxValue;

		for (int i = 0; i < width; i++) {
			for (int j = 0; j < height; j++) {

				noiseValues [i, j] = 0f;

				float tempA = amplitude;
				float tempF = frequency;

				for (int k = 0; k < octaves; k++) {

					noiseValues [i, j] += Mathf.PerlinNoise ((i + seed) / (float)width * frequency, (j + seed) / (float)height * frequency) * amplitude;
					frequency *= lacunarity;
					amplitude *= persistance;
				}

				amplitude = tempA;
				frequency = tempF;

				if (noiseValues [i, j] > max)
					max = noiseValues [i, j];

				if (noiseValues [i, j] < min)
					min = noiseValues [i, j];
			}
		}
		for (int i = 0; i < width; i++) {
			for (int j = 0; j < height; j++) {

				noiseValues [i, j] = Mathf.InverseLerp (max, min, noiseValues [i, j]);

			}
		}
		return noiseValues;
	}
}

And in my main class the map is created with the following function:

float[,] noiseValues = noise.GetNoiseValues (width, height);

		for (int x = 0; x < Width; x++) {
			for (int y = 0; y < Height; y++) {
				tiles[x,y] = new Tile(x, y);
                if (noiseValues[x,y]  < 0.2f) {
					tiles [x, y].Type = TileType.Shallow;
				} else if (noiseValues[x,y]  < 0.4f) {
					tiles [x, y].Type = TileType.Grass;
				} else if (noiseValues[x,y]  < 0.55f) {
					tiles [x, y].Type = TileType.Scrub;
				} else if (noiseValues[x,y]  < 0.6f) {
					tiles [x, y].Type = TileType.Marsh;
				} else if (noiseValues[x,y] < 1f) {
					tiles [x, y].Type = TileType.Bare;
				}
			}
		}

I’m very new to Unity and C#, I’ve done some reading around generating biomes but the math is somewhat above my head.
Is it possible to extend what I have to accomplish river generating?
Or could I use something like PathA* to join two points on a map to get the result I want?
Thanks!

I think A* would do perfectly the work for you

107808-pathfinding.jpg

to train PathFinding with A* i made a 2 grids with some wall (in your case the wall can be the black pixel )

Picture explication :

Black pixel = wall;
White pixel = free Space where the way can pass.
pink pixel = free Space visit by the A*
yellow pixel= Wall visit by the A*
blue/cyan pixel = Way to go from A to B
Red pixel = Point B : End
Green Pixel = Point A: Start