I was unable to find in Sense and Sensibility a moment when Sir John Middleton tells Elinor that he does not have a lot of money; however, perhaps you actually meant John Willoughby? Towards the end of the book, Willoughby comes to Elinor to explain his treatment of Marianne, and one of his reasons is lack of money. Below is an account of his explanations:
When Willoughby learns from Sir John Middleton that Marianne was dying of a fever at Cleveland, he rushes from town to try and make both Marianne and Elinor think "one degree less" badly of him (Ch. 44).
Willoughby confesses that at first when he met Marianne and her family, he had no other intention than flirting with her and enjoying her company, even if he saw that she sincerely loved him. His reason was that he already knew himself to be an extravagant man and knew that his estate at Combe Magna was not quite enough to pay for his expensive tastes and debts. Therefore, he also had to wait for his inheritance from Mrs. Smith; in the meantime, he decided that he must also marry a wealthy woman. Therefore, Willoughby tries to convince Elinor that his initial indifference to Marianne, though mean, selfish, and cruel, was due to his need for money.
However, Willoughby further explains that he did indeed fall in love with Marianne. He even states that he had made up his mind to forget about his need for money and propose to her when Mrs. Smith learned of his affair with Eliza. He never means to justify or excuse his affair with Eliza, but he does explain that Eliza was just as guilty of seduction as he was guilty of being a scoundrel. However, due to his affair with Eliza and his refusal to marry her because of his love for Marianne, Mrs. Smith disinherits him, leaving him in more need of money than ever. Hence, Willoughby made the resolution to give up the idea of marrying Marianne, leave for London immediately and court the wealthy Miss Grey instead.
Ultimately, Willoughby asks for compassion. Though he made his choices, he wants Elinor and Marianne to see how torturous it was for him. He wants them to see that he truly did love Marianne and meant to treat her rightly, but got caught up in his own other foolishness instead. As Willoughby phrases it, "If you can pity me, Miss Dashwood, pity my situation as it was then. With my head and heart full of your sister, I was forced to play the happy lover to another woman!" (Ch. 44).
Elinor does indeed feel compassion and pity for Willoughby, especially because his vanity and extravagance have made him "cold-hearted and selfish" when in reality he could be "open and honest," with an affectionate nature (Ch. 44).