Using React Native 0.41 (in March 2017), targeting iOS, I just found it as easy as:

<Image source={require('./myimage.png')} />

The image file must exist in the same folder as the .js file requiring it for "./" to work.

I didn't have to change anything in the XCode project. It just worked.

Note that the path seems to have to start with "./" or "../" and be full lower case. I'm not sure what all the restrictions are, but start simple and work forward.

Hope this helps someone, as many other answers here seem overly complex and full of (naughty) off-site links.

UPDATE: BTW - The official documentation for this is here: https://reactnative.dev/docs/images

If You have access to profile image in the render function then why don't you just past it to the function?

See example:

<View>

<TouchableOpacity onPress={() => profileImageFilter({uri: profileImage})}>
     <Image style={styles.profileimage} source={{uri: profileImage}}></Image>
</TouchableOpacity>

<TouchableOpacity onPress={() => profileImageFilter({uri: profileImage2})}>
     <Image style={styles.profileimage} source={{uri: profileImage2}}></Image>
</TouchableOpacity>

<TouchableOpacity onPress={() => profileImageFilter({uri: profileImage3})}>
     <Image style={styles.profileimage} source={{uri: profileImage3}}></Image>
</TouchableOpacity>

</View>

This is covered in the documentation under the section "Static Resources":

The only allowed way to refer to an image in the bundle is to literally write require('image!name-of-the-asset') in the source.

// GOOD
<Image source={require('image!my-icon')} />

// BAD
var icon = this.props.active ? 'my-icon-active' : 'my-icon-inactive';
<Image source={require('image!' + icon)} />

// GOOD
var icon = this.props.active ? require('image!my-icon-active') : require('image!my-icon-inactive');
<Image source={icon} />

However you also need to remember to add your images to an xcassets bundle in your app in Xcode, though it seems from your comment you've done that already.

http://facebook.github.io/react-native/docs/image.html#adding-static-resources-to-your-app-using-images-xcassets