Any op amp that includes V+ in it's common mode range (ie rail to rail input range)can be used to monitor a high side current sense resistor and to compare the voltage drop with a high side referenced voltage.
A current mirror is often used in this specific applicatiopn to "reflect" a high side referenced voltage to ground (usually with amplification). Specialist ICs are available to do this task but a pair of well-enough matched transistors can be used for the purpose. 
High side voltage sensing with opamps:
If using non rail-to-rail opamps the voltages on either side of the reference resistor may be divided down with a resistive divider to get the voltages within the available range.
Here is an example from fig 3 here - Planet Analog: Understand low-side vs. high-side current sensing
They divide the voltages before and after Vsense (Vcommon and say Vs) by 10:1 (9K:1k) but in doing so also reduce the delta voltage across the sense resistor by 10. Better where the absolute voltage allows is to divide by as little as possible. eg if the opamp will allow Vin to be within 1.5V of V+ and a 12V supply is used then Vin max = 12-1.5V = 10.5V or 10.5/12 = 0.875 of V+. To err on the safe side limit Vin max to say 2/3 = 0.66 of Vsupply or 8V in this case.
Here set R1 = 10k say and R2 = 20K or 22k using E12 resistor values. They show amplifier A2 with a gain of 250 - rather more than you'd hope to need in most cases.  Amplifier A1 needs to have a low enough input offset voltage to deal with the delta V experienced. If you set V_Rsense max = 0.5V and want 1% resolution then 1% of Vsense_max = 0.5V x 1% = 5 mV. A1 now sees 2/3 of that due tpo the resistor dividers or about 3 mV per 1% step in Vsense. Most common op-amps (evenones with otherwise quite good specs) can have Vinput_offset of more than 3 mV - but there are also many available with far far less. 

Application note from MSU using TI ICs
He uses eg the TI INA138 current sense IC - datasheet here
Gains of 1 to 100 from Rsense to Rl may be obrained with a maximum drop across across Rsense of 0.5V (lower OK). 
From the data sheet


Examples of dedicated ICs:
LTC6101
Other LT related solutions
A zillion DIY solutions
ADP122 datasheet