Let's assume the cost price (CP) of the article is x.
According to the question, the shopkeeper offers a 10% discount on the article, so the selling price (SP) becomes 90% of the CP.
Also, the shopkeeper still makes a profit of 20%, which means the SP is 120% of the CP.

**Calculating Selling Price (SP)**
SP = 120% of CP
SP = (120/100) * CP
SP = (6/5) * CP

**Calculating Discounted Selling Price (SP)**
SP = 90% of CP
SP = (90/100) * CP
SP = (9/10) * CP

Now, we can equate the two expressions for the selling price:
(6/5) * CP = (9/10) * CP

**Solving for CP**
(6/5) * CP = (9/10) * CP
6 * 10 = 9 * 5
60 = 45

This equation is not true, which means our assumption that the cost price is x is incorrect. Therefore, we need to find the correct value of CP.

We know that the article is marked at 500 rupees, and the shopkeeper offers a 10% discount. Therefore, the selling price after the discount is 90% of the marked price.

**Calculating Selling Price after Discount (SP)**
SP = 90% of marked price
SP = (90/100) * 500
SP = 450

We also know that the shopkeeper makes a profit of 20%, which means the selling price is 120% of the cost price.

**Calculating Cost Price (CP)**
CP = 120% of SP
CP = (120/100) * 450
CP = 540

Therefore, the cost price of the article is 540 rupees.

Now, let's check if the shopkeeper can offer a 10% discount and still make a profit of 20%.

**Calculating Selling Price after Discount (SP)**
SP = 90% of CP
SP = (90/100) * 540
SP = 486

**Calculating Profit**
Profit = SP - CP
Profit = 486 - 540
Profit = -54

We can see that the shopkeeper incurs a loss of 54 rupees, which contradicts the information given in the question. Hence, the given scenario is not possible.

Therefore, option D (375) cannot be the correct answer.